Calculus Examples

$x^{2} (x - 1) (x + 2)$

Step 1

Write $x^{2} (x - 1) (x + 2)$ as a function.

$f (x) = x^{2} (x - 1) (x + 2)$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} (x - 1)$ and $g (x) = x + 2$ .

$x^{2} (x - 1) \frac{d}{d x} [x + 2] + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$x^{2} (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (x - 1) (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x^{2} (x - 1) (1 + 0) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $1$ and $0$ .

$x^{2} (x - 1) \cdot 1 + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.2.4.2

Multiply $x^{2}$ by $1$ .

$x^{2} (x - 1) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x - 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (1 + 0) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

$x^{2} (x - 1) + (x + 2) (x^{2} \cdot 1 + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4.4.2

Multiply $x^{2}$ by $1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + (x - 1) \frac{d}{d x} [x^{2}])$

Step 2.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + (x - 1) (2 x))$

Step 2.4.6

Move $2$ to the left of $x - 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + 2 \cdot (x - 1) x)$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + 2 (x - 1) x)$

Step 2.5.2

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + (2 x + 2 \cdot - 1) x)$

Step 2.5.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + 2 x \cdot x + 2 \cdot - 1 x)$

Step 2.5.4

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) x^{2} + (x + 2) (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 2.5.5

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + (x + 2) (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 2.5.6

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 2.5.7

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8

Combine terms.

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Step 2.5.8.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.5.8.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.5.8.1.1.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.2

Move $- 1$ to the left of $x^{2}$ .

$x^{3} - 1 \cdot x^{2} + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$x^{3} - x^{2} + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.5.8.4.1

Multiply $x$ by $x^{2}$ .

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Step 2.5.8.4.1.1

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{1} x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{1 + 2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.4.2

Add $1$ and $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.5

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 (x^{1} x)) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.6

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 (x^{1} x^{1})) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x^{1 + 1}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.8

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x^{2}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.9

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 (x^{1} x^{2}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{1 + 2} + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.11

Add $1$ and $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.12

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 (x^{1} x)) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.13

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 (x^{1} x^{1})) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x^{1 + 1}) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.15

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x^{2}) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.16

Multiply $2$ by $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.17

Multiply $2$ by $- 1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} + x (- 2 x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.18

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 (x^{1} x) + 2 (2 \cdot - 1 x)$

Step 2.5.8.19

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 (x^{1} x^{1}) + 2 (2 \cdot - 1 x)$

Step 2.5.8.20

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{1 + 1} + 2 (2 \cdot - 1 x)$

Step 2.5.8.21

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} + 2 (2 \cdot - 1 x)$

Step 2.5.8.22

Multiply $2$ by $- 1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} + 2 (- 2 x)$

Step 2.5.8.23

Multiply $- 2$ by $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} - 4 x$

Step 2.5.8.24

Subtract $2 x^{2}$ from $4 x^{2}$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 x^{2} - 4 x$

Step 2.5.8.25

Add $x^{3}$ and $2 x^{3}$ .

$x^{3} - x^{2} + 3 x^{3} + 2 x^{2} + 2 x^{2} - 4 x$

Step 2.5.8.26

Add $2 x^{2}$ and $2 x^{2}$ .

$x^{3} - x^{2} + 3 x^{3} + 4 x^{2} - 4 x$

Step 2.5.8.27

Add $x^{3}$ and $3 x^{3}$ .

$4 x^{3} - x^{2} + 4 x^{2} - 4 x$

Step 2.5.8.28

Add $- x^{2}$ and $4 x^{2}$ .

$4 x^{3} + 3 x^{2} - 4 x$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $4 x^{3} + 3 x^{2} - 4 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 4 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 x)$

Step 3.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 x)$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 x)$

Step 3.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 x)$

Step 3.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4 x)$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 12 x^{2} + 3 (2 x) + \frac{d}{d x} (- 4 x)$

Step 3.3.3

Multiply $2$ by $3$ .

$f'' (x) = 12 x^{2} + 6 x + \frac{d}{d x} (- 4 x)$

Step 3.4

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 3.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} + 6 x - 4 \frac{d}{d x} x$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 12 x^{2} + 6 x - 4 \cdot 1$

Step 3.4.3

Multiply $- 4$ by $1$ .

$f'' (x) = 12 x^{2} + 6 x - 4$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$4 x^{3} + 3 x^{2} - 4 x = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} (x - 1)$ and $g (x) = x + 2$ .

$x^{2} (x - 1) \frac{d}{d x} [x + 2] + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$x^{2} (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (x - 1) (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x^{2} (x - 1) (1 + 0) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.2.4

Simplify the expression.

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Step 5.1.2.4.1

Add $1$ and $0$ .

$x^{2} (x - 1) \cdot 1 + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.2.4.2

Multiply $x^{2}$ by $1$ .

$x^{2} (x - 1) + (x + 2) \frac{d}{d x} [x^{2} (x - 1)]$

Step 5.1.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x - 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4

Differentiate.

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Step 5.1.4.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{2} (x - 1) + (x + 2) (x^{2} (1 + 0) + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4.4

Simplify the expression.

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Step 5.1.4.4.1

Add $1$ and $0$ .

$x^{2} (x - 1) + (x + 2) (x^{2} \cdot 1 + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4.4.2

Multiply $x^{2}$ by $1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + (x - 1) \frac{d}{d x} [x^{2}])$

Step 5.1.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + (x - 1) (2 x))$

Step 5.1.4.6

Move $2$ to the left of $x - 1$ .

$x^{2} (x - 1) + (x + 2) (x^{2} + 2 \cdot (x - 1) x)$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + 2 (x - 1) x)$

Step 5.1.5.2

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + (2 x + 2 \cdot - 1) x)$

Step 5.1.5.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) (x^{2} + 2 x \cdot x + 2 \cdot - 1 x)$

Step 5.1.5.4

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + (x + 2) x^{2} + (x + 2) (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 5.1.5.5

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + (x + 2) (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 5.1.5.6

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + (x + 2) (2 \cdot - 1 x)$

Step 5.1.5.7

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8

Combine terms.

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Step 5.1.5.8.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.1.5.8.1.1

Multiply $x^{2}$ by $x$ .

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Step 5.1.5.8.1.1.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 1 + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.2

Move $- 1$ to the left of $x^{2}$ .

$x^{3} - 1 \cdot x^{2} + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$x^{3} - x^{2} + x \cdot x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1.5.8.4.1

Multiply $x$ by $x^{2}$ .

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Step 5.1.5.8.4.1.1

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{1} x^{2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{1 + 2} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.4.2

Add $1$ and $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x \cdot x) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.5

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 (x^{1} x)) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.6

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 (x^{1} x^{1})) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x^{1 + 1}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.8

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + x (2 x^{2}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.9

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 (x^{1} x^{2}) + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{1 + 2} + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.11

Add $1$ and $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x \cdot x) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.12

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 (x^{1} x)) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.13

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 (x^{1} x^{1})) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x^{1 + 1}) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.15

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 (2 x^{2}) + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.16

Multiply $2$ by $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} + x (2 \cdot - 1 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.17

Multiply $2$ by $- 1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} + x (- 2 x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.18

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 (x^{1} x) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.19

Raise $x$ to the power of $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 (x^{1} x^{1}) + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.20

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{1 + 1} + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.21

Add $1$ and $1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} + 2 (2 \cdot - 1 x)$

Step 5.1.5.8.22

Multiply $2$ by $- 1$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} + 2 (- 2 x)$

Step 5.1.5.8.23

Multiply $- 2$ by $2$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 4 x^{2} - 2 x^{2} - 4 x$

Step 5.1.5.8.24

Subtract $2 x^{2}$ from $4 x^{2}$ .

$x^{3} - x^{2} + x^{3} + 2 x^{2} + 2 x^{3} + 2 x^{2} - 4 x$

Step 5.1.5.8.25

Add $x^{3}$ and $2 x^{3}$ .

$x^{3} - x^{2} + 3 x^{3} + 2 x^{2} + 2 x^{2} - 4 x$

Step 5.1.5.8.26

Add $2 x^{2}$ and $2 x^{2}$ .

$x^{3} - x^{2} + 3 x^{3} + 4 x^{2} - 4 x$

Step 5.1.5.8.27

Add $x^{3}$ and $3 x^{3}$ .

$4 x^{3} - x^{2} + 4 x^{2} - 4 x$

Step 5.1.5.8.28

Add $- x^{2}$ and $4 x^{2}$ .

$f' (x) = 4 x^{3} + 3 x^{2} - 4 x$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} + 3 x^{2} - 4 x$ .

$4 x^{3} + 3 x^{2} - 4 x$

Step 6

Set the first derivative equal to $0$ then solve the equation $4 x^{3} + 3 x^{2} - 4 x = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$4 x^{3} + 3 x^{2} - 4 x = 0$

Step 6.2

Factor $x$ out of $4 x^{3} + 3 x^{2} - 4 x$ .

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Step 6.2.1

Factor $x$ out of $4 x^{3}$ .

$x (4 x^{2}) + 3 x^{2} - 4 x = 0$

Step 6.2.2

Factor $x$ out of $3 x^{2}$ .

$x (4 x^{2}) + x (3 x) - 4 x = 0$

Step 6.2.3

Factor $x$ out of $- 4 x$ .

$x (4 x^{2}) + x (3 x) + x \cdot - 4 = 0$

Step 6.2.4

Factor $x$ out of $x (4 x^{2}) + x (3 x)$ .

$x (4 x^{2} + 3 x) + x \cdot - 4 = 0$

Step 6.2.5

Factor $x$ out of $x (4 x^{2} + 3 x) + x \cdot - 4$ .

$x (4 x^{2} + 3 x - 4) = 0$

Step 6.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x^{2} + 3 x - 4 = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

Set $4 x^{2} + 3 x - 4$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $4 x^{2} + 3 x - 4$ equal to $0$ .

$4 x^{2} + 3 x - 4 = 0$

Step 6.5.2

Solve $4 x^{2} + 3 x - 4 = 0$ for $x$ .

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Step 6.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.5.2.2

Substitute the values $a = 4$ , $b = 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (4 \cdot - 4)}}{2 \cdot 4}$

Step 6.5.2.3

Simplify.

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Step 6.5.2.3.1

Simplify the numerator.

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Step 6.5.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.3.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 6.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.3.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 6.5.2.3.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 4}$

Step 6.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{73}}{8}$

Step 6.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.5.2.4.1

Simplify the numerator.

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Step 6.5.2.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.4.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 6.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.4.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 6.5.2.4.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 4}$

Step 6.5.2.4.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{73}}{8}$

Step 6.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{73}}{8}$

Step 6.5.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{73}}{8}$

Step 6.5.2.4.5

Factor $- 1$ out of $\sqrt{73}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{73})}{8}$

Step 6.5.2.4.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{73})$ .

$x = \frac{- 1 (3 - \sqrt{73})}{8}$

Step 6.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{73}}{8}$

Step 6.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.5.2.5.1

Simplify the numerator.

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Step 6.5.2.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.5.1.2

Multiply $- 4 \cdot 4 \cdot - 4$ .

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Step 6.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 6.5.2.5.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 6.5.2.5.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 4}$

Step 6.5.2.5.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{73}}{8}$

Step 6.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{73}}{8}$

Step 6.5.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{73}}{8}$

Step 6.5.2.5.5

Factor $- 1$ out of $- \sqrt{73}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{73})}{8}$

Step 6.5.2.5.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{73})$ .

$x = \frac{- 1 (3 + \sqrt{73})}{8}$

Step 6.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{73}}{8}$

Step 6.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{73}}{8}, - \frac{3 + \sqrt{73}}{8}$

Step 6.6

The final solution is all the values that make $x (4 x^{2} + 3 x - 4) = 0$ true.

$x = 0, - \frac{3 - \sqrt{73}}{8}, - \frac{3 + \sqrt{73}}{8}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = 0, - \frac{3 - \sqrt{73}}{8}, - \frac{3 + \sqrt{73}}{8}$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(0)}^{2} + 6 (0) - 4$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raising $0$ to any positive power yields $0$ .

$12 \cdot 0 + 6 (0) - 4$

Step 10.1.2

Multiply $12$ by $0$ .

$0 + 6 (0) - 4$

Step 10.1.3

Multiply $6$ by $0$ .

$0 + 0 - 4$

Step 10.2

Simplify by adding and subtracting.

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Step 10.2.1

Add $0$ and $0$ .

$0 - 4$

Step 10.2.2

Subtract $4$ from $0$ .

$- 4$

Step 11

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 12

Find the y-value when $x = 0$ .

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Step 12.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} ((0) - 1) ((0) + 2)$

Step 12.2

Simplify the result.

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Step 12.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 ((0) - 1) ((0) + 2)$

Step 12.2.2

Subtract $1$ from $0$ .

$f (0) = 0 \cdot (- 1 ((0) + 2))$

Step 12.2.3

Multiply $0$ by $- 1$ .

$f (0) = 0 ((0) + 2)$

Step 12.2.4

Add $0$ and $2$ .

$f (0) = 0 \cdot 2$

Step 12.2.5

Multiply $0$ by $2$ .

$f (0) = 0$

Step 12.2.6

The final answer is $0$ .

$y = 0$

Step 13

Evaluate the second derivative at $x = - \frac{3 - \sqrt{73}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- \frac{3 - \sqrt{73}}{8})}^{2} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 14.1.1.1

Apply the product rule to $- \frac{3 - \sqrt{73}}{8}$ .

$12 ({(- 1)}^{2} {(\frac{3 - \sqrt{73}}{8})}^{2}) + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.1.2

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$12 ({(- 1)}^{2} \frac{{(3 - \sqrt{73})}^{2}}{8^{2}}) + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.2

Raise $- 1$ to the power of $2$ .

$12 (1 \frac{{(3 - \sqrt{73})}^{2}}{8^{2}}) + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.3

Multiply $\frac{{(3 - \sqrt{73})}^{2}}{8^{2}}$ by $1$ .

$12 \frac{{(3 - \sqrt{73})}^{2}}{8^{2}} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.4

Raise $8$ to the power of $2$ .

$12 \frac{{(3 - \sqrt{73})}^{2}}{64} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.5

Cancel the common factor of $4$ .

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Step 14.1.5.1

Factor $4$ out of $12$ .

$4 (3) \frac{{(3 - \sqrt{73})}^{2}}{64} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.5.2

Factor $4$ out of $64$ .

$4 \cdot 3 \frac{{(3 - \sqrt{73})}^{2}}{4 \cdot 16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.5.3

Cancel the common factor.

$4 \cdot 3 \frac{{(3 - \sqrt{73})}^{2}}{4 \cdot 16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.5.4

Rewrite the expression.

$3 \frac{{(3 - \sqrt{73})}^{2}}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.6

Combine $3$ and $\frac{{(3 - \sqrt{73})}^{2}}{16}$ .

$\frac{3 {(3 - \sqrt{73})}^{2}}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.7

Rewrite ${(3 - \sqrt{73})}^{2}$ as $(3 - \sqrt{73}) (3 - \sqrt{73})$ .

$\frac{3 ((3 - \sqrt{73}) (3 - \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.8

Expand $(3 - \sqrt{73}) (3 - \sqrt{73})$ using the FOIL Method.

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Step 14.1.8.1

Apply the distributive property.

$\frac{3 (3 (3 - \sqrt{73}) - \sqrt{73} (3 - \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.8.2

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} (3 - \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.8.3

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9

Simplify and combine like terms.

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Step 14.1.9.1

Simplify each term.

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Step 14.1.9.1.1

Multiply $3$ by $3$ .

$\frac{3 (9 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.2

Multiply $- 1$ by $3$ .

$\frac{3 (9 - 3 \sqrt{73} - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.3

Multiply $3$ by $- 1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} - \sqrt{73} (- \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4

Multiply $- \sqrt{73} (- \sqrt{73})$ .

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Step 14.1.9.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 1 \sqrt{73} \sqrt{73})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4.2

Multiply $\sqrt{73}$ by $1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1} \sqrt{73})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4.4

Raise $\sqrt{73}$ to the power of $1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1} {\sqrt{73}}^{1})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1 + 1})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.4.6

Add $1$ and $1$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{2})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 14.1.9.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {(73^{\frac{1}{2}})}^{2})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{1}{2} \cdot 2})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5.4

Cancel the common factor of $2$ .

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Step 14.1.9.1.5.4.1

Cancel the common factor.

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5.4.2

Rewrite the expression.

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{1})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.1.5.5

Evaluate the exponent.

$\frac{3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73)}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.2

Add $9$ and $73$ .

$\frac{3 (82 - 3 \sqrt{73} - 3 \sqrt{73})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.9.3

Subtract $3 \sqrt{73}$ from $- 3 \sqrt{73}$ .

$\frac{3 (82 - 6 \sqrt{73})}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.10

Cancel the common factor of $82 - 6 \sqrt{73}$ and $16$ .

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Step 14.1.10.1

Factor $2$ out of $3 (82 - 6 \sqrt{73})$ .

$\frac{2 (3 (41 - 3 \sqrt{73}))}{16} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.10.2

Cancel the common factors.

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Step 14.1.10.2.1

Factor $2$ out of $16$ .

$\frac{2 (3 (41 - 3 \sqrt{73}))}{2 (8)} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.10.2.2

Cancel the common factor.

$\frac{2 (3 (41 - 3 \sqrt{73}))}{2 \cdot 8} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.10.2.3

Rewrite the expression.

$\frac{3 (41 - 3 \sqrt{73})}{8} + 6 (- \frac{3 - \sqrt{73}}{8}) - 4$

Step 14.1.11

Cancel the common factor of $2$ .

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Step 14.1.11.1

Move the leading negative in $- \frac{3 - \sqrt{73}}{8}$ into the numerator.

$\frac{3 (41 - 3 \sqrt{73})}{8} + 6 \frac{- (3 - \sqrt{73})}{8} - 4$

Step 14.1.11.2

Factor $2$ out of $6$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} + 2 (3) \frac{- (3 - \sqrt{73})}{8} - 4$

Step 14.1.11.3

Factor $2$ out of $8$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{- (3 - \sqrt{73})}{2 \cdot 4} - 4$

Step 14.1.11.4

Cancel the common factor.

$\frac{3 (41 - 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{- (3 - \sqrt{73})}{2 \cdot 4} - 4$

Step 14.1.11.5

Rewrite the expression.

$\frac{3 (41 - 3 \sqrt{73})}{8} + 3 \frac{- (3 - \sqrt{73})}{4} - 4$

Step 14.1.12

Combine $3$ and $\frac{- (3 - \sqrt{73})}{4}$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (- (3 - \sqrt{73}))}{4} - 4$

Step 14.1.13

Multiply $- 1$ by $3$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} + \frac{- 3 (3 - \sqrt{73})}{4} - 4$

Step 14.1.14

Move the negative in front of the fraction.

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73})}{4} - 4$

Step 14.2

Find the common denominator.

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Step 14.2.1

Multiply $\frac{3 (3 - \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - (\frac{3 (3 - \sqrt{73})}{4} \cdot \frac{2}{2}) - 4$

Step 14.2.2

Multiply $\frac{3 (3 - \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} - 4$

Step 14.2.3

Write $- 4$ as a fraction with denominator $1$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4}{1}$

Step 14.2.4

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4}{1} \cdot \frac{8}{8}$

Step 14.2.5

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4 \cdot 8}{8}$

Step 14.2.6

Reorder the factors of $4 \cdot 2$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{2 \cdot 4} + \frac{- 4 \cdot 8}{8}$

Step 14.2.7

Multiply $2$ by $4$ .

$\frac{3 (41 - 3 \sqrt{73})}{8} - \frac{3 (3 - \sqrt{73}) \cdot 2}{8} + \frac{- 4 \cdot 8}{8}$

Step 14.3

Combine the numerators over the common denominator.

$\frac{3 (41 - 3 \sqrt{73}) - 3 (3 - \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4

Simplify each term.

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Step 14.4.1

Apply the distributive property.

$\frac{3 \cdot 41 + 3 (- 3 \sqrt{73}) - 3 (3 - \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.2

Multiply $3$ by $41$ .

$\frac{123 + 3 (- 3 \sqrt{73}) - 3 (3 - \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.3

Multiply $- 3$ by $3$ .

$\frac{123 - 9 \sqrt{73} - 3 (3 - \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.4

Apply the distributive property.

$\frac{123 - 9 \sqrt{73} + (- 3 \cdot 3 - 3 (- \sqrt{73})) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.5

Multiply $- 3$ by $3$ .

$\frac{123 - 9 \sqrt{73} + (- 9 - 3 (- \sqrt{73})) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.6

Multiply $- 1$ by $- 3$ .

$\frac{123 - 9 \sqrt{73} + (- 9 + 3 \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.7

Apply the distributive property.

$\frac{123 - 9 \sqrt{73} - 9 \cdot 2 + 3 \sqrt{73} \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.8

Multiply $- 9$ by $2$ .

$\frac{123 - 9 \sqrt{73} - 18 + 3 \sqrt{73} \cdot 2 - 4 \cdot 8}{8}$

Step 14.4.9

Multiply $2$ by $3$ .

$\frac{123 - 9 \sqrt{73} - 18 + 6 \sqrt{73} - 4 \cdot 8}{8}$

Step 14.4.10

Multiply $- 4$ by $8$ .

$\frac{123 - 9 \sqrt{73} - 18 + 6 \sqrt{73} - 32}{8}$

Step 14.5

Simplify by adding terms.

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Step 14.5.1

Subtract $18$ from $123$ .

$\frac{105 - 9 \sqrt{73} + 6 \sqrt{73} - 32}{8}$

Step 14.5.2

Subtract $32$ from $105$ .

$\frac{73 - 9 \sqrt{73} + 6 \sqrt{73}}{8}$

Step 14.5.3

Add $- 9 \sqrt{73}$ and $6 \sqrt{73}$ .

$\frac{73 - 3 \sqrt{73}}{8}$

Step 15

$x = - \frac{3 - \sqrt{73}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{3 - \sqrt{73}}{8}$ is a local minimum

Step 16

Find the y-value when $x = - \frac{3 - \sqrt{73}}{8}$ .

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Step 16.1

Replace the variable $x$ with $- \frac{3 - \sqrt{73}}{8}$ in the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = {(- \frac{3 - \sqrt{73}}{8})}^{2} ((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2

Simplify the result.

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Step 16.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 16.2.1.1

Apply the product rule to $- \frac{3 - \sqrt{73}}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = {(- 1)}^{2} {(\frac{3 - \sqrt{73}}{8})}^{2} ((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.1.2

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = {(- 1)}^{2} (\frac{{(3 - \sqrt{73})}^{2}}{8^{2}}) \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.2

Simplify the expression.

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Step 16.2.2.1

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 - \sqrt{73}}{8}) = 1 (\frac{{(3 - \sqrt{73})}^{2}}{8^{2}}) \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.2.2

Multiply $\frac{{(3 - \sqrt{73})}^{2}}{8^{2}}$ by $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{{(3 - \sqrt{73})}^{2}}{8^{2}} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.2.3

Raise $8$ to the power of $2$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{{(3 - \sqrt{73})}^{2}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.2.4

Rewrite ${(3 - \sqrt{73})}^{2}$ as $(3 - \sqrt{73}) (3 - \sqrt{73})$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{(3 - \sqrt{73}) (3 - \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.3

Expand $(3 - \sqrt{73}) (3 - \sqrt{73})$ using the FOIL Method.

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Step 16.2.3.1

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{3 (3 - \sqrt{73}) - \sqrt{73} (3 - \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.3.2

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} (3 - \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.3.3

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4

Simplify and combine like terms.

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Step 16.2.4.1

Simplify each term.

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Step 16.2.4.1.1

Multiply $3$ by $3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.2

Multiply $- 1$ by $3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.3

Multiply $3$ by $- 1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} - \sqrt{73} (- \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4

Multiply $- \sqrt{73} (- \sqrt{73})$ .

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Step 16.2.4.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 1 \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4.2

Multiply $\sqrt{73}$ by $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4.4

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1 + 1}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.4.6

Add $1$ and $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{2}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 16.2.4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {(73^{\frac{1}{2}})}^{2}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{1}{2} \cdot 2}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5.4

Cancel the common factor of $2$ .

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Step 16.2.4.1.5.4.1

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5.4.2

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.1.5.5

Evaluate the exponent.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.2

Add $9$ and $73$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{82 - 3 \sqrt{73} - 3 \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.4.3

Subtract $3 \sqrt{73}$ from $- 3 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{82 - 6 \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5

Cancel the common factor of $82 - 6 \sqrt{73}$ and $64$ .

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Step 16.2.5.1

Factor $2$ out of $82$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (41) - 6 \sqrt{73}}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5.2

Factor $2$ out of $- 6 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (41) + 2 (- 3 \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5.3

Factor $2$ out of $2 (41) + 2 (- 3 \sqrt{73})$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (41 - 3 \sqrt{73})}{64} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5.4

Cancel the common factors.

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Step 16.2.5.4.1

Factor $2$ out of $64$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (41 - 3 \sqrt{73})}{2 \cdot 32} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5.4.2

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (41 - 3 \sqrt{73})}{2 \cdot 32} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.5.4.3

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot (((- \frac{3 - \sqrt{73}}{8}) - 1) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot ((- \frac{3 - \sqrt{73}}{8} - 1 \cdot \frac{8}{8}) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.7

Combine fractions.

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Step 16.2.7.1

Combine $- 1$ and $\frac{8}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot ((- \frac{3 - \sqrt{73}}{8} + \frac{- 1 \cdot 8}{8}) ((- \frac{3 - \sqrt{73}}{8}) + 2))$

Step 16.2.7.2

Combine the numerators over the common denominator.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- (3 - \sqrt{73}) - 1 \cdot 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8

Simplify the numerator.

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Step 16.2.8.1

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 1 \cdot 3 + \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8.2

Multiply $- 1$ by $3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 3 + \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8.3

Multiply $- - \sqrt{73}$ .

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Step 16.2.8.3.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 3 + 1 \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8.3.2

Multiply $\sqrt{73}$ by $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 3 + \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8.4

Multiply $- 1$ by $8$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 3 + \sqrt{73} - 8}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.8.5

Subtract $8$ from $- 3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 11 + \sqrt{73}}{8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.9

Multiply $\frac{41 - 3 \sqrt{73}}{32} \cdot \frac{- 11 + \sqrt{73}}{8}$ .

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Step 16.2.9.1

Multiply $\frac{41 - 3 \sqrt{73}}{32}$ by $\frac{- 11 + \sqrt{73}}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{(41 - 3 \sqrt{73}) (- 11 + \sqrt{73})}{32 \cdot 8} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.9.2

Multiply $32$ by $8$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{(41 - 3 \sqrt{73}) (- 11 + \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.10

Expand $(41 - 3 \sqrt{73}) (- 11 + \sqrt{73})$ using the FOIL Method.

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Step 16.2.10.1

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 (- 11 + \sqrt{73}) - 3 \sqrt{73} (- 11 + \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.10.2

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 \cdot - 11 + 41 \sqrt{73} - 3 \sqrt{73} (- 11 + \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.10.3

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{41 \cdot - 11 + 41 \sqrt{73} - 3 \sqrt{73} \cdot - 11 - 3 \sqrt{73} \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11

Simplify and combine like terms.

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Step 16.2.11.1

Simplify each term.

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Step 16.2.11.1.1

Multiply $41$ by $- 11$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} - 3 \sqrt{73} \cdot - 11 - 3 \sqrt{73} \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.2

Multiply $- 11$ by $- 3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \sqrt{73} \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.3

Multiply $- 3 \sqrt{73} \sqrt{73}$ .

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Step 16.2.11.1.3.1

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 (\sqrt{73} \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.3.2

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 (\sqrt{73} \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 {\sqrt{73}}^{1 + 1}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.3.4

Add $1$ and $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 {\sqrt{73}}^{2}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 16.2.11.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 {(73^{\frac{1}{2}})}^{2}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \cdot 73^{\frac{1}{2} \cdot 2}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \cdot 73^{\frac{2}{2}}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4.4

Cancel the common factor of $2$ .

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Step 16.2.11.1.4.4.1

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \cdot 73^{\frac{2}{2}}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4.4.2

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \cdot 73}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.4.5

Evaluate the exponent.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 3 \cdot 73}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.1.5

Multiply $- 3$ by $73$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 451 + 41 \sqrt{73} + 33 \sqrt{73} - 219}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.2

Subtract $219$ from $- 451$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 670 + 41 \sqrt{73} + 33 \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.11.3

Add $41 \sqrt{73}$ and $33 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 670 + 74 \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12

Cancel the common factor of $- 670 + 74 \sqrt{73}$ and $256$ .

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Step 16.2.12.1

Factor $2$ out of $- 670$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 335) + 74 \sqrt{73}}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12.2

Factor $2$ out of $74 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 335) + 2 (37 \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12.3

Factor $2$ out of $2 (- 335) + 2 (37 \sqrt{73})$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 335 + 37 \sqrt{73})}{256} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12.4

Cancel the common factors.

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Step 16.2.12.4.1

Factor $2$ out of $256$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 335 + 37 \sqrt{73})}{2 \cdot 128} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12.4.2

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 335 + 37 \sqrt{73})}{2 \cdot 128} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.12.4.3

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot ((- \frac{3 - \sqrt{73}}{8}) + 2)$

Step 16.2.13

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot (- \frac{3 - \sqrt{73}}{8} + 2 \cdot \frac{8}{8})$

Step 16.2.14

Combine fractions.

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Step 16.2.14.1

Combine $2$ and $\frac{8}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot (- \frac{3 - \sqrt{73}}{8} + \frac{2 \cdot 8}{8})$

Step 16.2.14.2

Combine the numerators over the common denominator.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- (3 - \sqrt{73}) + 2 \cdot 8}{8}$

Step 16.2.15

Simplify the numerator.

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Step 16.2.15.1

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- 1 \cdot 3 + \sqrt{73} + 2 \cdot 8}{8}$

Step 16.2.15.2

Multiply $- 1$ by $3$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- 3 + \sqrt{73} + 2 \cdot 8}{8}$

Step 16.2.15.3

Multiply $- - \sqrt{73}$ .

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Step 16.2.15.3.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- 3 + 1 \sqrt{73} + 2 \cdot 8}{8}$

Step 16.2.15.3.2

Multiply $\sqrt{73}$ by $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- 3 + \sqrt{73} + 2 \cdot 8}{8}$

Step 16.2.15.4

Multiply $2$ by $8$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{- 3 + \sqrt{73} + 16}{8}$

Step 16.2.15.5

Add $- 3$ and $16$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{13 + \sqrt{73}}{8}$

Step 16.2.16

Multiply $\frac{- 335 + 37 \sqrt{73}}{128} \cdot \frac{13 + \sqrt{73}}{8}$ .

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Step 16.2.16.1

Multiply $\frac{- 335 + 37 \sqrt{73}}{128}$ by $\frac{13 + \sqrt{73}}{8}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{(- 335 + 37 \sqrt{73}) (13 + \sqrt{73})}{128 \cdot 8}$

Step 16.2.16.2

Multiply $128$ by $8$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{(- 335 + 37 \sqrt{73}) (13 + \sqrt{73})}{1024}$

Step 16.2.17

Expand $(- 335 + 37 \sqrt{73}) (13 + \sqrt{73})$ using the FOIL Method.

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Step 16.2.17.1

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 (13 + \sqrt{73}) + 37 \sqrt{73} (13 + \sqrt{73})}{1024}$

Step 16.2.17.2

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 \cdot 13 - 335 \sqrt{73} + 37 \sqrt{73} (13 + \sqrt{73})}{1024}$

Step 16.2.17.3

Apply the distributive property.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 335 \cdot 13 - 335 \sqrt{73} + 37 \sqrt{73} \cdot 13 + 37 \sqrt{73} \sqrt{73}}{1024}$

Step 16.2.18

Simplify and combine like terms.

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Step 16.2.18.1

Simplify each term.

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Step 16.2.18.1.1

Multiply $- 335$ by $13$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 37 \sqrt{73} \cdot 13 + 37 \sqrt{73} \sqrt{73}}{1024}$

Step 16.2.18.1.2

Multiply $13$ by $37$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \sqrt{73} \sqrt{73}}{1024}$

Step 16.2.18.1.3

Multiply $37 \sqrt{73} \sqrt{73}$ .

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Step 16.2.18.1.3.1

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 (\sqrt{73} \sqrt{73})}{1024}$

Step 16.2.18.1.3.2

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 (\sqrt{73} \sqrt{73})}{1024}$

Step 16.2.18.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 {\sqrt{73}}^{1 + 1}}{1024}$

Step 16.2.18.1.3.4

Add $1$ and $1$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 {\sqrt{73}}^{2}}{1024}$

Step 16.2.18.1.4

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 16.2.18.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 {(73^{\frac{1}{2}})}^{2}}{1024}$

Step 16.2.18.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \cdot 73^{\frac{1}{2} \cdot 2}}{1024}$

Step 16.2.18.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \cdot 73^{\frac{2}{2}}}{1024}$

Step 16.2.18.1.4.4

Cancel the common factor of $2$ .

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Step 16.2.18.1.4.4.1

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \cdot 73^{\frac{2}{2}}}{1024}$

Step 16.2.18.1.4.4.2

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \cdot 73}{1024}$

Step 16.2.18.1.4.5

Evaluate the exponent.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 37 \cdot 73}{1024}$

Step 16.2.18.1.5

Multiply $37$ by $73$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 4355 - 335 \sqrt{73} + 481 \sqrt{73} + 2701}{1024}$

Step 16.2.18.2

Add $- 4355$ and $2701$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 1654 - 335 \sqrt{73} + 481 \sqrt{73}}{1024}$

Step 16.2.18.3

Add $- 335 \sqrt{73}$ and $481 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 1654 + 146 \sqrt{73}}{1024}$

Step 16.2.19

Cancel the common factor of $- 1654 + 146 \sqrt{73}$ and $1024$ .

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Step 16.2.19.1

Factor $2$ out of $- 1654$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 827) + 146 \sqrt{73}}{1024}$

Step 16.2.19.2

Factor $2$ out of $146 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 827) + 2 (73 \sqrt{73})}{1024}$

Step 16.2.19.3

Factor $2$ out of $2 (- 827) + 2 (73 \sqrt{73})$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 827 + 73 \sqrt{73})}{1024}$

Step 16.2.19.4

Cancel the common factors.

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Step 16.2.19.4.1

Factor $2$ out of $1024$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 827 + 73 \sqrt{73})}{2 \cdot 512}$

Step 16.2.19.4.2

Cancel the common factor.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{2 (- 827 + 73 \sqrt{73})}{2 \cdot 512}$

Step 16.2.19.4.3

Rewrite the expression.

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 827 + 73 \sqrt{73}}{512}$

Step 16.2.20

Rewrite $- 827$ as $- 1 (827)$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 1 \cdot 827 + 73 \sqrt{73}}{512}$

Step 16.2.21

Factor $- 1$ out of $73 \sqrt{73}$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 1 \cdot 827 - (- 73 \sqrt{73})}{512}$

Step 16.2.22

Factor $- 1$ out of $- 1 (827) - (- 73 \sqrt{73})$ .

$f (- \frac{3 - \sqrt{73}}{8}) = \frac{- 1 (827 - 73 \sqrt{73})}{512}$

Step 16.2.23

Move the negative in front of the fraction.

$f (- \frac{3 - \sqrt{73}}{8}) = - \frac{827 - 73 \sqrt{73}}{512}$

Step 16.2.24

The final answer is $- \frac{827 - 73 \sqrt{73}}{512}$ .

$y = - \frac{827 - 73 \sqrt{73}}{512}$

Step 17

Evaluate the second derivative at $x = - \frac{3 + \sqrt{73}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- \frac{3 + \sqrt{73}}{8})}^{2} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18

Evaluate the second derivative.

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Step 18.1

Simplify each term.

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Step 18.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 18.1.1.1

Apply the product rule to $- \frac{3 + \sqrt{73}}{8}$ .

$12 ({(- 1)}^{2} {(\frac{3 + \sqrt{73}}{8})}^{2}) + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.1.2

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$12 ({(- 1)}^{2} \frac{{(3 + \sqrt{73})}^{2}}{8^{2}}) + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.2

Raise $- 1$ to the power of $2$ .

$12 (1 \frac{{(3 + \sqrt{73})}^{2}}{8^{2}}) + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.3

Multiply $\frac{{(3 + \sqrt{73})}^{2}}{8^{2}}$ by $1$ .

$12 \frac{{(3 + \sqrt{73})}^{2}}{8^{2}} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.4

Raise $8$ to the power of $2$ .

$12 \frac{{(3 + \sqrt{73})}^{2}}{64} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.5

Cancel the common factor of $4$ .

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Step 18.1.5.1

Factor $4$ out of $12$ .

$4 (3) \frac{{(3 + \sqrt{73})}^{2}}{64} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.5.2

Factor $4$ out of $64$ .

$4 \cdot 3 \frac{{(3 + \sqrt{73})}^{2}}{4 \cdot 16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.5.3

Cancel the common factor.

$4 \cdot 3 \frac{{(3 + \sqrt{73})}^{2}}{4 \cdot 16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.5.4

Rewrite the expression.

$3 \frac{{(3 + \sqrt{73})}^{2}}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.6

Combine $3$ and $\frac{{(3 + \sqrt{73})}^{2}}{16}$ .

$\frac{3 {(3 + \sqrt{73})}^{2}}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.7

Rewrite ${(3 + \sqrt{73})}^{2}$ as $(3 + \sqrt{73}) (3 + \sqrt{73})$ .

$\frac{3 ((3 + \sqrt{73}) (3 + \sqrt{73}))}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.8

Expand $(3 + \sqrt{73}) (3 + \sqrt{73})$ using the FOIL Method.

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Step 18.1.8.1

Apply the distributive property.

$\frac{3 (3 (3 + \sqrt{73}) + \sqrt{73} (3 + \sqrt{73}))}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.8.2

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} (3 + \sqrt{73}))}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.8.3

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9

Simplify and combine like terms.

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Step 18.1.9.1

Simplify each term.

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Step 18.1.9.1.1

Multiply $3$ by $3$ .

$\frac{3 (9 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.1.2

Move $3$ to the left of $\sqrt{73}$ .

$\frac{3 (9 + 3 \sqrt{73} + 3 \cdot \sqrt{73} + \sqrt{73} \sqrt{73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.1.3

Combine using the product rule for radicals.

$\frac{3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73 \cdot 73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.1.4

Multiply $73$ by $73$ .

$\frac{3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{5329})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.1.5

Rewrite $5329$ as $73^{2}$ .

$\frac{3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73^{2}})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + 73)}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.2

Add $9$ and $73$ .

$\frac{3 (82 + 3 \sqrt{73} + 3 \sqrt{73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.9.3

Add $3 \sqrt{73}$ and $3 \sqrt{73}$ .

$\frac{3 (82 + 6 \sqrt{73})}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.10

Cancel the common factor of $82 + 6 \sqrt{73}$ and $16$ .

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Step 18.1.10.1

Factor $2$ out of $3 (82 + 6 \sqrt{73})$ .

$\frac{2 (3 (41 + 3 \sqrt{73}))}{16} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.10.2

Cancel the common factors.

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Step 18.1.10.2.1

Factor $2$ out of $16$ .

$\frac{2 (3 (41 + 3 \sqrt{73}))}{2 (8)} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.10.2.2

Cancel the common factor.

$\frac{2 (3 (41 + 3 \sqrt{73}))}{2 \cdot 8} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.10.2.3

Rewrite the expression.

$\frac{3 (41 + 3 \sqrt{73})}{8} + 6 (- \frac{3 + \sqrt{73}}{8}) - 4$

Step 18.1.11

Cancel the common factor of $2$ .

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Step 18.1.11.1

Move the leading negative in $- \frac{3 + \sqrt{73}}{8}$ into the numerator.

$\frac{3 (41 + 3 \sqrt{73})}{8} + 6 \frac{- (3 + \sqrt{73})}{8} - 4$

Step 18.1.11.2

Factor $2$ out of $6$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} + 2 (3) \frac{- (3 + \sqrt{73})}{8} - 4$

Step 18.1.11.3

Factor $2$ out of $8$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{- (3 + \sqrt{73})}{2 \cdot 4} - 4$

Step 18.1.11.4

Cancel the common factor.

$\frac{3 (41 + 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{- (3 + \sqrt{73})}{2 \cdot 4} - 4$

Step 18.1.11.5

Rewrite the expression.

$\frac{3 (41 + 3 \sqrt{73})}{8} + 3 \frac{- (3 + \sqrt{73})}{4} - 4$

Step 18.1.12

Combine $3$ and $\frac{- (3 + \sqrt{73})}{4}$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (- (3 + \sqrt{73}))}{4} - 4$

Step 18.1.13

Multiply $- 1$ by $3$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} + \frac{- 3 (3 + \sqrt{73})}{4} - 4$

Step 18.1.14

Move the negative in front of the fraction.

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73})}{4} - 4$

Step 18.2

Find the common denominator.

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Step 18.2.1

Multiply $\frac{3 (3 + \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - (\frac{3 (3 + \sqrt{73})}{4} \cdot \frac{2}{2}) - 4$

Step 18.2.2

Multiply $\frac{3 (3 + \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} - 4$

Step 18.2.3

Write $- 4$ as a fraction with denominator $1$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4}{1}$

Step 18.2.4

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4}{1} \cdot \frac{8}{8}$

Step 18.2.5

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{- 4 \cdot 8}{8}$

Step 18.2.6

Reorder the factors of $4 \cdot 2$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{2 \cdot 4} + \frac{- 4 \cdot 8}{8}$

Step 18.2.7

Multiply $2$ by $4$ .

$\frac{3 (41 + 3 \sqrt{73})}{8} - \frac{3 (3 + \sqrt{73}) \cdot 2}{8} + \frac{- 4 \cdot 8}{8}$

Step 18.3

Combine the numerators over the common denominator.

$\frac{3 (41 + 3 \sqrt{73}) - 3 (3 + \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4

Simplify each term.

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Step 18.4.1

Apply the distributive property.

$\frac{3 \cdot 41 + 3 (3 \sqrt{73}) - 3 (3 + \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.2

Multiply $3$ by $41$ .

$\frac{123 + 3 (3 \sqrt{73}) - 3 (3 + \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.3

Multiply $3$ by $3$ .

$\frac{123 + 9 \sqrt{73} - 3 (3 + \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.4

Apply the distributive property.

$\frac{123 + 9 \sqrt{73} + (- 3 \cdot 3 - 3 \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.5

Multiply $- 3$ by $3$ .

$\frac{123 + 9 \sqrt{73} + (- 9 - 3 \sqrt{73}) \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.6

Apply the distributive property.

$\frac{123 + 9 \sqrt{73} - 9 \cdot 2 - 3 \sqrt{73} \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.7

Multiply $- 9$ by $2$ .

$\frac{123 + 9 \sqrt{73} - 18 - 3 \sqrt{73} \cdot 2 - 4 \cdot 8}{8}$

Step 18.4.8

Multiply $2$ by $- 3$ .

$\frac{123 + 9 \sqrt{73} - 18 - 6 \sqrt{73} - 4 \cdot 8}{8}$

Step 18.4.9

Multiply $- 4$ by $8$ .

$\frac{123 + 9 \sqrt{73} - 18 - 6 \sqrt{73} - 32}{8}$

Step 18.5

Simplify by adding terms.

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Step 18.5.1

Subtract $18$ from $123$ .

$\frac{105 + 9 \sqrt{73} - 6 \sqrt{73} - 32}{8}$

Step 18.5.2

Subtract $32$ from $105$ .

$\frac{73 + 9 \sqrt{73} - 6 \sqrt{73}}{8}$

Step 18.5.3

Subtract $6 \sqrt{73}$ from $9 \sqrt{73}$ .

$\frac{73 + 3 \sqrt{73}}{8}$

Step 19

$x = - \frac{3 + \sqrt{73}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{3 + \sqrt{73}}{8}$ is a local minimum

Step 20

Find the y-value when $x = - \frac{3 + \sqrt{73}}{8}$ .

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Step 20.1

Replace the variable $x$ with $- \frac{3 + \sqrt{73}}{8}$ in the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = {(- \frac{3 + \sqrt{73}}{8})}^{2} ((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2

Simplify the result.

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Step 20.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 20.2.1.1

Apply the product rule to $- \frac{3 + \sqrt{73}}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = {(- 1)}^{2} {(\frac{3 + \sqrt{73}}{8})}^{2} ((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.1.2

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = {(- 1)}^{2} (\frac{{(3 + \sqrt{73})}^{2}}{8^{2}}) \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.2

Simplify the expression.

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Step 20.2.2.1

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 + \sqrt{73}}{8}) = 1 (\frac{{(3 + \sqrt{73})}^{2}}{8^{2}}) \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.2.2

Multiply $\frac{{(3 + \sqrt{73})}^{2}}{8^{2}}$ by $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{{(3 + \sqrt{73})}^{2}}{8^{2}} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.2.3

Raise $8$ to the power of $2$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{{(3 + \sqrt{73})}^{2}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.2.4

Rewrite ${(3 + \sqrt{73})}^{2}$ as $(3 + \sqrt{73}) (3 + \sqrt{73})$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{(3 + \sqrt{73}) (3 + \sqrt{73})}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.3

Expand $(3 + \sqrt{73}) (3 + \sqrt{73})$ using the FOIL Method.

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Step 20.2.3.1

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{3 (3 + \sqrt{73}) + \sqrt{73} (3 + \sqrt{73})}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.3.2

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} (3 + \sqrt{73})}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.3.3

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4

Simplify and combine like terms.

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Step 20.2.4.1

Simplify each term.

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Step 20.2.4.1.1

Multiply $3$ by $3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.1.2

Move $3$ to the left of $\sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + 3 \cdot \sqrt{73} + \sqrt{73} \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.1.3

Combine using the product rule for radicals.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73 \cdot 73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.1.4

Multiply $73$ by $73$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{5329}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.1.5

Rewrite $5329$ as $73^{2}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73^{2}}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + 73}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.2

Add $9$ and $73$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{82 + 3 \sqrt{73} + 3 \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.4.3

Add $3 \sqrt{73}$ and $3 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{82 + 6 \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5

Cancel the common factor of $82 + 6 \sqrt{73}$ and $64$ .

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Step 20.2.5.1

Factor $2$ out of $82$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (41) + 6 \sqrt{73}}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5.2

Factor $2$ out of $6 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (41) + 2 (3 \sqrt{73})}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5.3

Factor $2$ out of $2 (41) + 2 (3 \sqrt{73})$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (41 + 3 \sqrt{73})}{64} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5.4

Cancel the common factors.

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Step 20.2.5.4.1

Factor $2$ out of $64$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (41 + 3 \sqrt{73})}{2 \cdot 32} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5.4.2

Cancel the common factor.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (41 + 3 \sqrt{73})}{2 \cdot 32} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.5.4.3

Rewrite the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot (((- \frac{3 + \sqrt{73}}{8}) - 1) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot ((- \frac{3 + \sqrt{73}}{8} - 1 \cdot \frac{8}{8}) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.7

Combine fractions.

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Step 20.2.7.1

Combine $- 1$ and $\frac{8}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot ((- \frac{3 + \sqrt{73}}{8} + \frac{- 1 \cdot 8}{8}) ((- \frac{3 + \sqrt{73}}{8}) + 2))$

Step 20.2.7.2

Combine the numerators over the common denominator.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- (3 + \sqrt{73}) - 1 \cdot 8}{8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.8

Simplify the numerator.

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Step 20.2.8.1

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- 1 \cdot 3 - \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.8.2

Multiply $- 1$ by $3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- 3 - \sqrt{73} - 1 \cdot 8}{8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.8.3

Multiply $- 1$ by $8$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- 3 - \sqrt{73} - 8}{8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.8.4

Subtract $8$ from $- 3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- 11 - \sqrt{73}}{8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.9

Multiply $\frac{41 + 3 \sqrt{73}}{32} \cdot \frac{- 11 - \sqrt{73}}{8}$ .

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Step 20.2.9.1

Multiply $\frac{41 + 3 \sqrt{73}}{32}$ by $\frac{- 11 - \sqrt{73}}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{(41 + 3 \sqrt{73}) (- 11 - \sqrt{73})}{32 \cdot 8} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.9.2

Multiply $32$ by $8$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{(41 + 3 \sqrt{73}) (- 11 - \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.10

Expand $(41 + 3 \sqrt{73}) (- 11 - \sqrt{73})$ using the FOIL Method.

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Step 20.2.10.1

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 (- 11 - \sqrt{73}) + 3 \sqrt{73} (- 11 - \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.10.2

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 \cdot - 11 + 41 (- \sqrt{73}) + 3 \sqrt{73} (- 11 - \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.10.3

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{41 \cdot - 11 + 41 (- \sqrt{73}) + 3 \sqrt{73} \cdot - 11 + 3 \sqrt{73} (- \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11

Simplify and combine like terms.

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Step 20.2.11.1

Simplify each term.

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Step 20.2.11.1.1

Multiply $41$ by $- 11$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 + 41 (- \sqrt{73}) + 3 \sqrt{73} \cdot - 11 + 3 \sqrt{73} (- \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.2

Multiply $- 1$ by $41$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} + 3 \sqrt{73} \cdot - 11 + 3 \sqrt{73} (- \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.3

Multiply $- 11$ by $3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} + 3 \sqrt{73} (- \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.4

Multiply $3 \sqrt{73} (- \sqrt{73})$ .

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Step 20.2.11.1.4.1

Multiply $- 1$ by $3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \sqrt{73} \sqrt{73}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.4.2

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 (\sqrt{73} \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 (\sqrt{73} \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 {\sqrt{73}}^{1 + 1}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.4.5

Add $1$ and $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 {\sqrt{73}}^{2}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 20.2.11.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 {(73^{\frac{1}{2}})}^{2}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \cdot 73^{\frac{1}{2} \cdot 2}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \cdot 73^{\frac{2}{2}}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5.4

Cancel the common factor of $2$ .

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Step 20.2.11.1.5.4.1

Cancel the common factor.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \cdot 73^{\frac{2}{2}}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5.4.2

Rewrite the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \cdot 73}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.5.5

Evaluate the exponent.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 3 \cdot 73}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.1.6

Multiply $- 3$ by $73$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 451 - 41 \sqrt{73} - 33 \sqrt{73} - 219}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.2

Subtract $219$ from $- 451$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 670 - 41 \sqrt{73} - 33 \sqrt{73}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.11.3

Subtract $33 \sqrt{73}$ from $- 41 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 670 - 74 \sqrt{73}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12

Cancel the common factor of $- 670 - 74 \sqrt{73}$ and $256$ .

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Step 20.2.12.1

Factor $2$ out of $- 670$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 335) - 74 \sqrt{73}}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12.2

Factor $2$ out of $- 74 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 335) + 2 (- 37 \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12.3

Factor $2$ out of $2 (- 335) + 2 (- 37 \sqrt{73})$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 335 - 37 \sqrt{73})}{256} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12.4

Cancel the common factors.

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Step 20.2.12.4.1

Factor $2$ out of $256$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 335 - 37 \sqrt{73})}{2 \cdot 128} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12.4.2

Cancel the common factor.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 335 - 37 \sqrt{73})}{2 \cdot 128} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.12.4.3

Rewrite the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot ((- \frac{3 + \sqrt{73}}{8}) + 2)$

Step 20.2.13

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot (- \frac{3 + \sqrt{73}}{8} + 2 \cdot \frac{8}{8})$

Step 20.2.14

Combine fractions.

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Step 20.2.14.1

Combine $2$ and $\frac{8}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot (- \frac{3 + \sqrt{73}}{8} + \frac{2 \cdot 8}{8})$

Step 20.2.14.2

Combine the numerators over the common denominator.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{- (3 + \sqrt{73}) + 2 \cdot 8}{8}$

Step 20.2.15

Simplify the numerator.

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Step 20.2.15.1

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{- 1 \cdot 3 - \sqrt{73} + 2 \cdot 8}{8}$

Step 20.2.15.2

Multiply $- 1$ by $3$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{- 3 - \sqrt{73} + 2 \cdot 8}{8}$

Step 20.2.15.3

Multiply $2$ by $8$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{- 3 - \sqrt{73} + 16}{8}$

Step 20.2.15.4

Add $- 3$ and $16$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{13 - \sqrt{73}}{8}$

Step 20.2.16

Multiply $\frac{- 335 - 37 \sqrt{73}}{128} \cdot \frac{13 - \sqrt{73}}{8}$ .

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Step 20.2.16.1

Multiply $\frac{- 335 - 37 \sqrt{73}}{128}$ by $\frac{13 - \sqrt{73}}{8}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{(- 335 - 37 \sqrt{73}) (13 - \sqrt{73})}{128 \cdot 8}$

Step 20.2.16.2

Multiply $128$ by $8$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{(- 335 - 37 \sqrt{73}) (13 - \sqrt{73})}{1024}$

Step 20.2.17

Expand $(- 335 - 37 \sqrt{73}) (13 - \sqrt{73})$ using the FOIL Method.

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Step 20.2.17.1

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 (13 - \sqrt{73}) - 37 \sqrt{73} (13 - \sqrt{73})}{1024}$

Step 20.2.17.2

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 \cdot 13 - 335 (- \sqrt{73}) - 37 \sqrt{73} (13 - \sqrt{73})}{1024}$

Step 20.2.17.3

Apply the distributive property.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 335 \cdot 13 - 335 (- \sqrt{73}) - 37 \sqrt{73} \cdot 13 - 37 \sqrt{73} (- \sqrt{73})}{1024}$

Step 20.2.18

Simplify and combine like terms.

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Step 20.2.18.1

Simplify each term.

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Step 20.2.18.1.1

Multiply $- 335$ by $13$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 - 335 (- \sqrt{73}) - 37 \sqrt{73} \cdot 13 - 37 \sqrt{73} (- \sqrt{73})}{1024}$

Step 20.2.18.1.2

Multiply $- 1$ by $- 335$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 37 \sqrt{73} \cdot 13 - 37 \sqrt{73} (- \sqrt{73})}{1024}$

Step 20.2.18.1.3

Multiply $13$ by $- 37$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} - 37 \sqrt{73} (- \sqrt{73})}{1024}$

Step 20.2.18.1.4

Multiply $- 37 \sqrt{73} (- \sqrt{73})$ .

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Step 20.2.18.1.4.1

Multiply $- 1$ by $- 37$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \sqrt{73} \sqrt{73}}{1024}$

Step 20.2.18.1.4.2

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 (\sqrt{73} \sqrt{73})}{1024}$

Step 20.2.18.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 (\sqrt{73} \sqrt{73})}{1024}$

Step 20.2.18.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 {\sqrt{73}}^{1 + 1}}{1024}$

Step 20.2.18.1.4.5

Add $1$ and $1$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 {\sqrt{73}}^{2}}{1024}$

Step 20.2.18.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 20.2.18.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 {(73^{\frac{1}{2}})}^{2}}{1024}$

Step 20.2.18.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \cdot 73^{\frac{1}{2} \cdot 2}}{1024}$

Step 20.2.18.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \cdot 73^{\frac{2}{2}}}{1024}$

Step 20.2.18.1.5.4

Cancel the common factor of $2$ .

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Step 20.2.18.1.5.4.1

Cancel the common factor.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \cdot 73^{\frac{2}{2}}}{1024}$

Step 20.2.18.1.5.4.2

Rewrite the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \cdot 73}{1024}$

Step 20.2.18.1.5.5

Evaluate the exponent.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 37 \cdot 73}{1024}$

Step 20.2.18.1.6

Multiply $37$ by $73$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 4355 + 335 \sqrt{73} - 481 \sqrt{73} + 2701}{1024}$

Step 20.2.18.2

Add $- 4355$ and $2701$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 1654 + 335 \sqrt{73} - 481 \sqrt{73}}{1024}$

Step 20.2.18.3

Subtract $481 \sqrt{73}$ from $335 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 1654 - 146 \sqrt{73}}{1024}$

Step 20.2.19

Cancel the common factor of $- 1654 - 146 \sqrt{73}$ and $1024$ .

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Step 20.2.19.1

Factor $2$ out of $- 1654$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 827) - 146 \sqrt{73}}{1024}$

Step 20.2.19.2

Factor $2$ out of $- 146 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 827) + 2 (- 73 \sqrt{73})}{1024}$

Step 20.2.19.3

Factor $2$ out of $2 (- 827) + 2 (- 73 \sqrt{73})$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 827 - 73 \sqrt{73})}{1024}$

Step 20.2.19.4

Cancel the common factors.

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Step 20.2.19.4.1

Factor $2$ out of $1024$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 827 - 73 \sqrt{73})}{2 \cdot 512}$

Step 20.2.19.4.2

Cancel the common factor.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{2 (- 827 - 73 \sqrt{73})}{2 \cdot 512}$

Step 20.2.19.4.3

Rewrite the expression.

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 827 - 73 \sqrt{73}}{512}$

Step 20.2.20

Rewrite $- 827$ as $- 1 (827)$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 1 \cdot 827 - 73 \sqrt{73}}{512}$

Step 20.2.21

Factor $- 1$ out of $- 73 \sqrt{73}$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 1 \cdot 827 - (73 \sqrt{73})}{512}$

Step 20.2.22

Factor $- 1$ out of $- 1 (827) - (73 \sqrt{73})$ .

$f (- \frac{3 + \sqrt{73}}{8}) = \frac{- 1 (827 + 73 \sqrt{73})}{512}$

Step 20.2.23

Move the negative in front of the fraction.

$f (- \frac{3 + \sqrt{73}}{8}) = - \frac{827 + 73 \sqrt{73}}{512}$

Step 20.2.24

The final answer is $- \frac{827 + 73 \sqrt{73}}{512}$ .

$y = - \frac{827 + 73 \sqrt{73}}{512}$

Step 21

These are the local extrema for $f (x) = x^{2} (x - 1) (x + 2)$ .

$(0, 0)$ is a local maxima

$(- \frac{3 - \sqrt{73}}{8}, - \frac{827 - 73 \sqrt{73}}{512})$ is a local minima

$(- \frac{3 + \sqrt{73}}{8}, - \frac{827 + 73 \sqrt{73}}{512})$ is a local minima

Step 22