Calculus Examples

$f (x) = 0.25 (x + 4) (x + 1) (x - 2)$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $0.25$ is constant with respect to $x$ , the derivative of $0.25 (x + 4) (x + 1) (x - 2)$ with respect to $x$ is $0.25 \frac{d}{d x} [((x + 4) (x + 1)) (x - 2)]$ .

$0.25 \frac{d}{d x} [((x + 4) (x + 1)) (x - 2)]$

Step 1.2

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 $f (x) = (x + 4) (x + 1)$ and $g (x) = x - 2$ .

$0.25 ((x + 4) (x + 1) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.3

Differentiate.

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Step 1.3.1

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

$0.25 ((x + 4) (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.3.2

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

$0.25 ((x + 4) (x + 1) (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.3.3

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

$0.25 ((x + 4) (x + 1) (1 + 0) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) \cdot 1 + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.3.4.2

Multiply $x + 4$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 1.4

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 $f (x) = x + 4$ and $g (x) = x + 1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) \frac{d}{d x} [x + 1] + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5

Differentiate.

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Step 1.5.1

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5.2

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (1 + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5.3

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (1 + 0) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5.4

Simplify the expression.

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Step 1.5.4.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) \cdot 1 + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5.4.2

Multiply $x + 4$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) \frac{d}{d x} [x + 4]))$

Step 1.5.5

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

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [4])))$

Step 1.5.6

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

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (1 + \frac{d}{d x} [4])))$

Step 1.5.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (1 + 0)))$

Step 1.5.8

Simplify by adding terms.

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Step 1.5.8.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) \cdot 1))$

Step 1.5.8.2

Multiply $x + 1$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + x + 1))$

Step 1.5.8.3

Add $x$ and $x$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (2 x + 4 + 1))$

Step 1.5.8.4

Add $4$ and $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (2 x + 5))$

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

$0.25 ((x + 4) (x + 1)) + 0.25 ((x - 2) (2 x + 5))$

Step 1.6.2

Apply the distributive property.

$0.25 ((x + 4) x + (x + 4) \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 1.6.3

Apply the distributive property.

$0.25 (x \cdot x + 4 x + (x + 4) \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 1.6.4

Apply the distributive property.

$0.25 (x \cdot x + 4 x + x \cdot 1 + 4 \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 1.6.5

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 1.6.6

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 ((x - 2) (2 x) + (x - 2) \cdot 5)$

Step 1.6.7

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x) - 2 (2 x) + (x - 2) \cdot 5)$

Step 1.6.8

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x) - 2 (2 x) + x \cdot 5 - 2 \cdot 5)$

Step 1.6.9

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10

Combine terms.

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Step 1.6.10.1

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

$0.25 (x^{1} x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.2

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

$0.25 (x^{1} x^{1}) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.3

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

$0.25 x^{1 + 1} + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.4

Add $1$ and $1$ .

$0.25 x^{2} + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.5

Multiply $4$ by $0.25$ .

$0.25 x^{2} + 1 x + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.6

Multiply $x$ by $1$ .

$0.25 x^{2} + x + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.7

Multiply $x$ by $1$ .

$0.25 x^{2} + x + 0.25 x + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.8

Multiply $4$ by $1$ .

$0.25 x^{2} + x + 0.25 x + 0.25 \cdot 4 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.9

Multiply $0.25$ by $4$ .

$0.25 x^{2} + x + 0.25 x + 1 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.10

Add $x$ and $0.25 x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.11

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 (x^{1} x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.12

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 (x^{1} x^{1})) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.13

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 x^{1 + 1}) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.14

Add $1$ and $1$ .

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 x^{2}) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.15

Multiply $2$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.16

Multiply $2$ by $- 2$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 (- 4 x) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.17

Multiply $- 4$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.18

Move $5$ to the left of $x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 0.25 (5 \cdot x) + 0.25 (- 2 \cdot 5)$

Step 1.6.10.19

Multiply $5$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x + 0.25 (- 2 \cdot 5)$

Step 1.6.10.20

Multiply $- 2$ by $5$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x + 0.25 \cdot - 10$

Step 1.6.10.21

Multiply $0.25$ by $- 10$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x - 2.5$

Step 1.6.10.22

Add $- x$ and $1.25 x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 x - 2.5$

Step 1.6.10.23

Add $0.25 x^{2}$ and $0.5 x^{2}$ .

$0.75 x^{2} + 1.25 x + 1 + 0.25 x - 2.5$

Step 1.6.10.24

Add $1.25 x$ and $0.25 x$ .

$0.75 x^{2} + 1.5 x + 1 - 2.5$

Step 1.6.10.25

Subtract $2.5$ from $1$ .

$0.75 x^{2} + 1.5 x - 1.5$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $0.75 x^{2} + 1.5 x - 1.5$ with respect to $x$ is $\frac{d}{d x} [0.75 x^{2}] + \frac{d}{d x} [1.5 x] + \frac{d}{d x} [- 1.5]$ .

$f'' (x) = \frac{d}{d x} (0.75 x^{2}) + \frac{d}{d x} (1.5 x) + \frac{d}{d x} (- 1.5)$

Step 2.2

Evaluate $\frac{d}{d x} [0.75 x^{2}]$ .

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Step 2.2.1

Since $0.75$ is constant with respect to $x$ , the derivative of $0.75 x^{2}$ with respect to $x$ is $0.75 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 0.75 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (1.5 x) + \frac{d}{d x} (- 1.5)$

Step 2.2.2

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

$f'' (x) = 0.75 (2 x) + \frac{d}{d x} (1.5 x) + \frac{d}{d x} (- 1.5)$

Step 2.2.3

Multiply $2$ by $0.75$ .

$f'' (x) = 1.5 x + \frac{d}{d x} (1.5 x) + \frac{d}{d x} (- 1.5)$

Step 2.3

Evaluate $\frac{d}{d x} [1.5 x]$ .

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Step 2.3.1

Since $1.5$ is constant with respect to $x$ , the derivative of $1.5 x$ with respect to $x$ is $1.5 \frac{d}{d x} [x]$ .

$f'' (x) = 1.5 x + 1.5 \frac{d}{d x} (x) + \frac{d}{d x} (- 1.5)$

Step 2.3.2

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

$f'' (x) = 1.5 x + 1.5 \cdot 1 + \frac{d}{d x} (- 1.5)$

Step 2.3.3

Multiply $1.5$ by $1$ .

$f'' (x) = 1.5 x + 1.5 + \frac{d}{d x} (- 1.5)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 1.5$ is constant with respect to $x$ , the derivative of $- 1.5$ with respect to $x$ is $0$ .

$f'' (x) = 1.5 x + 1.5 + 0$

Step 2.4.2

Add $1.5 x + 1.5$ and $0$ .

$f'' (x) = 1.5 x + 1.5$

Step 3

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

$0.75 x^{2} + 1.5 x - 1.5 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $0.25$ is constant with respect to $x$ , the derivative of $0.25 (x + 4) (x + 1) (x - 2)$ with respect to $x$ is $0.25 \frac{d}{d x} [((x + 4) (x + 1)) (x - 2)]$ .

$0.25 \frac{d}{d x} [((x + 4) (x + 1)) (x - 2)]$

Step 4.1.2

$0.25 ((x + 4) (x + 1) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

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

$0.25 ((x + 4) (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.3.2

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

$0.25 ((x + 4) (x + 1) (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.3.3

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

$0.25 ((x + 4) (x + 1) (1 + 0) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.3.4

Simplify the expression.

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Step 4.1.3.4.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) \cdot 1 + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.3.4.2

Multiply $x + 4$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) \frac{d}{d x} [(x + 4) (x + 1)])$

Step 4.1.4

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 $f (x) = x + 4$ and $g (x) = x + 1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) \frac{d}{d x} [x + 1] + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5

Differentiate.

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Step 4.1.5.1

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5.2

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (1 + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5.3

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

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) (1 + 0) + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5.4

Simplify the expression.

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Step 4.1.5.4.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) ((x + 4) \cdot 1 + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5.4.2

Multiply $x + 4$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) \frac{d}{d x} [x + 4]))$

Step 4.1.5.5

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

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [4])))$

Step 4.1.5.6

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

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (1 + \frac{d}{d x} [4])))$

Step 4.1.5.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) (1 + 0)))$

Step 4.1.5.8

Simplify by adding terms.

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Step 4.1.5.8.1

Add $1$ and $0$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + (x + 1) \cdot 1))$

Step 4.1.5.8.2

Multiply $x + 1$ by $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (x + 4 + x + 1))$

Step 4.1.5.8.3

Add $x$ and $x$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (2 x + 4 + 1))$

Step 4.1.5.8.4

Add $4$ and $1$ .

$0.25 ((x + 4) (x + 1) + (x - 2) (2 x + 5))$

Step 4.1.6

Simplify.

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Step 4.1.6.1

Apply the distributive property.

$0.25 ((x + 4) (x + 1)) + 0.25 ((x - 2) (2 x + 5))$

Step 4.1.6.2

Apply the distributive property.

$0.25 ((x + 4) x + (x + 4) \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 4.1.6.3

Apply the distributive property.

$0.25 (x \cdot x + 4 x + (x + 4) \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 4.1.6.4

Apply the distributive property.

$0.25 (x \cdot x + 4 x + x \cdot 1 + 4 \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 4.1.6.5

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 ((x - 2) (2 x + 5))$

Step 4.1.6.6

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 ((x - 2) (2 x) + (x - 2) \cdot 5)$

Step 4.1.6.7

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x) - 2 (2 x) + (x - 2) \cdot 5)$

Step 4.1.6.8

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x) - 2 (2 x) + x \cdot 5 - 2 \cdot 5)$

Step 4.1.6.9

Apply the distributive property.

$0.25 (x \cdot x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10

Combine terms.

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Step 4.1.6.10.1

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

$0.25 (x^{1} x) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.2

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

$0.25 (x^{1} x^{1}) + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.3

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

$0.25 x^{1 + 1} + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.4

Add $1$ and $1$ .

$0.25 x^{2} + 0.25 (4 x) + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.5

Multiply $4$ by $0.25$ .

$0.25 x^{2} + 1 x + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.6

Multiply $x$ by $1$ .

$0.25 x^{2} + x + 0.25 (x \cdot 1) + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.7

Multiply $x$ by $1$ .

$0.25 x^{2} + x + 0.25 x + 0.25 (4 \cdot 1) + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.8

Multiply $4$ by $1$ .

$0.25 x^{2} + x + 0.25 x + 0.25 \cdot 4 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.9

Multiply $0.25$ by $4$ .

$0.25 x^{2} + x + 0.25 x + 1 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.10

Add $x$ and $0.25 x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.25 (x (2 x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.11

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 (x^{1} x)) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.12

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 (x^{1} x^{1})) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.13

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

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 x^{1 + 1}) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.14

Add $1$ and $1$ .

$0.25 x^{2} + 1.25 x + 1 + 0.25 (2 x^{2}) + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.15

Multiply $2$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 (- 2 (2 x)) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.16

Multiply $2$ by $- 2$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 (- 4 x) + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.17

Multiply $- 4$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 0.25 (x \cdot 5) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.18

Move $5$ to the left of $x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 0.25 (5 \cdot x) + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.19

Multiply $5$ by $0.25$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x + 0.25 (- 2 \cdot 5)$

Step 4.1.6.10.20

Multiply $- 2$ by $5$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x + 0.25 \cdot - 10$

Step 4.1.6.10.21

Multiply $0.25$ by $- 10$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} - x + 1.25 x - 2.5$

Step 4.1.6.10.22

Add $- x$ and $1.25 x$ .

$0.25 x^{2} + 1.25 x + 1 + 0.5 x^{2} + 0.25 x - 2.5$

Step 4.1.6.10.23

Add $0.25 x^{2}$ and $0.5 x^{2}$ .

$0.75 x^{2} + 1.25 x + 1 + 0.25 x - 2.5$

Step 4.1.6.10.24

Add $1.25 x$ and $0.25 x$ .

$0.75 x^{2} + 1.5 x + 1 - 2.5$

Step 4.1.6.10.25

Subtract $2.5$ from $1$ .

$f' (x) = 0.75 x^{2} + 1.5 x - 1.5$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $0.75 x^{2} + 1.5 x - 1.5$ .

$0.75 x^{2} + 1.5 x - 1.5$

Step 5

Set the first derivative equal to $0$ then solve the equation $0.75 x^{2} + 1.5 x - 1.5 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$0.75 x^{2} + 1.5 x - 1.5 = 0$

Step 5.2

Factor $0.75$ out of $0.75 x^{2} + 1.5 x - 1.5$ .

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Step 5.2.1

Factor $0.75$ out of $0.75 x^{2}$ .

$0.75 (x^{2}) + 1.5 x - 1.5 = 0$

Step 5.2.2

Factor $0.75$ out of $1.5 x$ .

$0.75 (x^{2}) + 0.75 (2 x) - 1.5 = 0$

Step 5.2.3

Factor $0.75$ out of $- 1.5$ .

$0.75 x^{2} + 0.75 (2 x) + 0.75 \cdot - 2 = 0$

Step 5.2.4

Factor $0.75$ out of $0.75 x^{2} + 0.75 (2 x)$ .

$0.75 (x^{2} + 2 x) + 0.75 \cdot - 2 = 0$

Step 5.2.5

Factor $0.75$ out of $0.75 (x^{2} + 2 x) + 0.75 \cdot - 2$ .

$0.75 (x^{2} + 2 x - 2) = 0$

Step 5.3

Divide each term in $0.75 (x^{2} + 2 x - 2) = 0$ by $0.75$ and simplify.

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Step 5.3.1

Divide each term in $0.75 (x^{2} + 2 x - 2) = 0$ by $0.75$ .

$\frac{0.75 (x^{2} + 2 x - 2)}{0.75} = \frac{0}{0.75}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $0.75$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.75 (x^{2} + 2 x - 2)}{0.75} = \frac{0}{0.75}$

Step 5.3.2.1.2

Divide $x^{2} + 2 x - 2$ by $1$ .

$x^{2} + 2 x - 2 = \frac{0}{0.75}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $0$ by $0.75$ .

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

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

Substitute the values $a = 1$ , $b = 2$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 5.6.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 2}}{2 \cdot 1}$

Step 5.6.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 2 \pm \sqrt{4 + 8}}{2 \cdot 1}$

Step 5.6.1.3

Add $4$ and $8$ .

$x = \frac{- 2 \pm \sqrt{12}}{2 \cdot 1}$

Step 5.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.6.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 5.6.2

Multiply $2$ by $1$ .

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

Step 5.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{3}}{2}$ .

$x = - 1 \pm \sqrt{3}$

Step 5.7

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

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Step 5.7.1

Simplify the numerator.

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Step 5.7.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 5.7.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 5.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 2}}{2 \cdot 1}$

Step 5.7.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 2 \pm \sqrt{4 + 8}}{2 \cdot 1}$

Step 5.7.1.3

Add $4$ and $8$ .

$x = \frac{- 2 \pm \sqrt{12}}{2 \cdot 1}$

Step 5.7.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.7.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 5.7.2

Multiply $2$ by $1$ .

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

Step 5.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{3}}{2}$ .

$x = - 1 \pm \sqrt{3}$

Step 5.7.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{3}$

Step 5.8

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

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Step 5.8.1

Simplify the numerator.

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Step 5.8.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 5.8.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 5.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 2}}{2 \cdot 1}$

Step 5.8.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 2 \pm \sqrt{4 + 8}}{2 \cdot 1}$

Step 5.8.1.3

Add $4$ and $8$ .

$x = \frac{- 2 \pm \sqrt{12}}{2 \cdot 1}$

Step 5.8.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.8.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.8.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 5.8.2

Multiply $2$ by $1$ .

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

Step 5.8.3

Simplify $\frac{- 2 \pm 2 \sqrt{3}}{2}$ .

$x = - 1 \pm \sqrt{3}$

Step 5.8.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{3}$

Step 5.9

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{3}, - 1 - \sqrt{3}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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 7

Critical points to evaluate.

$x = - 1 + \sqrt{3}, - 1 - \sqrt{3}$

Step 8

Evaluate the second derivative at $x = - 1 + \sqrt{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$1.5 (- 1 + \sqrt{3}) + 1.5$

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $1.5$ by $- 1 + \sqrt{3}$ .

$1.09807621 + 1.5$

Step 9.2

Add $1.09807621$ and $1.5$ .

$2.59807621$

Step 10

$x = - 1 + \sqrt{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 1 + \sqrt{3}$ is a local minimum

Step 11

Find the y-value when $x = - 1 + \sqrt{3}$ .

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Step 11.1

Replace the variable $x$ with $- 1 + \sqrt{3}$ in the expression.

$f (- 1 + \sqrt{3}) = 0.25 ((- 1 + \sqrt{3}) + 4) ((- 1 + \sqrt{3}) + 1) ((- 1 + \sqrt{3}) - 2)$

Step 11.2

Simplify the result.

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Step 11.2.1

Add $- 1$ and $4$ .

$f (- 1 + \sqrt{3}) = 0.25 (3 + \sqrt{3}) ((- 1 + \sqrt{3}) + 1) ((- 1 + \sqrt{3}) - 2)$

Step 11.2.2

Multiply $0.25$ by $3 + \sqrt{3}$ .

$f (- 1 + \sqrt{3}) = 1.1830127 ((- 1 + \sqrt{3}) + 1) ((- 1 + \sqrt{3}) - 2)$

Step 11.2.3

Simplify by adding numbers.

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Step 11.2.3.1

Add $- 1$ and $1$ .

$f (- 1 + \sqrt{3}) = 1.1830127 (0 + \sqrt{3}) ((- 1 + \sqrt{3}) - 2)$

Step 11.2.3.2

Add $0$ and $\sqrt{3}$ .

$f (- 1 + \sqrt{3}) = 1.1830127 \sqrt{3} ((- 1 + \sqrt{3}) - 2)$

Step 11.2.4

Multiply $1.1830127$ by $\sqrt{3}$ .

$f (- 1 + \sqrt{3}) = 2.0490381 ((- 1 + \sqrt{3}) - 2)$

Step 11.2.5

Subtract $2$ from $- 1$ .

$f (- 1 + \sqrt{3}) = 2.0490381 (- 3 + \sqrt{3})$

Step 11.2.6

Multiply $2.0490381$ by $- 3 + \sqrt{3}$ .

$f (- 1 + \sqrt{3}) = - 2.59807621$

Step 11.2.7

The final answer is $- 2.59807621$ .

$y = - 2.59807621$

Step 12

Evaluate the second derivative at $x = - 1 - \sqrt{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$1.5 (- 1 - \sqrt{3}) + 1.5$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $1.5$ by $- 1 - \sqrt{3}$ .

$- 4.09807621 + 1.5$

Step 13.2

Add $- 4.09807621$ and $1.5$ .

$- 2.59807621$

Step 14

$x = - 1 - \sqrt{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 1 - \sqrt{3}$ is a local maximum

Step 15

Find the y-value when $x = - 1 - \sqrt{3}$ .

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Step 15.1

Replace the variable $x$ with $- 1 - \sqrt{3}$ in the expression.

$f (- 1 - \sqrt{3}) = 0.25 ((- 1 - \sqrt{3}) + 4) ((- 1 - \sqrt{3}) + 1) ((- 1 - \sqrt{3}) - 2)$

Step 15.2

Simplify the result.

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Step 15.2.1

Add $- 1$ and $4$ .

$f (- 1 - \sqrt{3}) = 0.25 (3 - \sqrt{3}) ((- 1 - \sqrt{3}) + 1) ((- 1 - \sqrt{3}) - 2)$

Step 15.2.2

Multiply $0.25$ by $3 - \sqrt{3}$ .

$f (- 1 - \sqrt{3}) = 0.31698729 ((- 1 - \sqrt{3}) + 1) ((- 1 - \sqrt{3}) - 2)$

Step 15.2.3

Simplify by adding numbers.

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Step 15.2.3.1

Add $- 1$ and $1$ .

$f (- 1 - \sqrt{3}) = 0.31698729 (0 - \sqrt{3}) ((- 1 - \sqrt{3}) - 2)$

Step 15.2.3.2

Subtract $\sqrt{3}$ from $0$ .

$f (- 1 - \sqrt{3}) = 0.31698729 (- \sqrt{3}) ((- 1 - \sqrt{3}) - 2)$

Step 15.2.4

Multiply $0.31698729 (- \sqrt{3})$ .

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Step 15.2.4.1

Multiply $- 1$ by $0.31698729$ .

$f (- 1 - \sqrt{3}) = - 0.31698729 \sqrt{3} ((- 1 - \sqrt{3}) - 2)$

Step 15.2.4.2

Multiply $- 0.31698729$ by $\sqrt{3}$ .

$f (- 1 - \sqrt{3}) = - 0.5490381 ((- 1 - \sqrt{3}) - 2)$

Step 15.2.5

Subtract $2$ from $- 1$ .

$f (- 1 - \sqrt{3}) = - 0.5490381 (- 3 - \sqrt{3})$

Step 15.2.6

Multiply $- 0.5490381$ by $- 3 - \sqrt{3}$ .

$f (- 1 - \sqrt{3}) = 2.59807621$

Step 15.2.7

The final answer is $2.59807621$ .

$y = 2.59807621$

Step 16

These are the local extrema for $f (x) = 0.25 (x + 4) (x + 1) (x - 2)$ .

$(- 1 + \sqrt{3}, - 2.59807621)$ is a local minima

$(- 1 - \sqrt{3}, 2.59807621)$ is a local maxima

Step 17