Calculus Examples

$f (x) = 24 x^{3} - 3 x^{4} \cdot (x^{3} (24 - 3 x))$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $24 x^{3} - 3 x^{4} \cdot (x^{3} (24 - 3 x))$ with respect to $x$ is $\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$ .

$\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 1.2

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

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Step 1.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$24 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 1.2.2

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

$24 (3 x^{2}) + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 1.2.3

Multiply $3$ by $24$ .

$72 x^{2} + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 1.3

Evaluate $\frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$ .

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Step 1.3.1

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

$72 x^{2} + \frac{d}{d x} [- 3 x^{3 + 4} \cdot (24 - 3 x)]$

Step 1.3.2

Add $3$ and $4$ .

$72 x^{2} + \frac{d}{d x} [- 3 x^{7} \cdot (24 - 3 x)]$

Step 1.3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{7} (24 - 3 x)$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{7} (24 - 3 x)]$ .

$72 x^{2} - 3 \frac{d}{d x} [x^{7} (24 - 3 x)]$

Step 1.3.4

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 $f (x) = x^{7}$ and $g (x) = 24 - 3 x$ .

$72 x^{2} - 3 (x^{7} \frac{d}{d x} [24 - 3 x] + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.5

By the Sum Rule, the derivative of $24 - 3 x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- 3 x]$ .

$72 x^{2} - 3 (x^{7} (\frac{d}{d x} [24] + \frac{d}{d x} [- 3 x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.6

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$72 x^{2} - 3 (x^{7} (0 + \frac{d}{d x} [- 3 x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.7

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

$72 x^{2} - 3 (x^{7} (0 - 3 \frac{d}{d x} [x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.8

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

$72 x^{2} - 3 (x^{7} (0 - 3 \cdot 1) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$72 x^{2} - 3 (x^{7} (0 - 3 \cdot 1) + (24 - 3 x) (7 x^{6}))$

Step 1.3.10

Multiply $- 3$ by $1$ .

$72 x^{2} - 3 (x^{7} (0 - 3) + (24 - 3 x) (7 x^{6}))$

Step 1.3.11

Subtract $3$ from $0$ .

$72 x^{2} - 3 (x^{7} \cdot - 3 + (24 - 3 x) (7 x^{6}))$

Step 1.3.12

Move $- 3$ to the left of $x^{7}$ .

$72 x^{2} - 3 (- 3 \cdot x^{7} + (24 - 3 x) (7 x^{6}))$

Step 1.3.13

Move $7$ to the left of $24 - 3 x$ .

$72 x^{2} - 3 (- 3 x^{7} + 7 (24 - 3 x) x^{6})$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7} + (7 \cdot 24 + 7 (- 3 x)) x^{6})$

Step 1.4.2

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7} + 7 \cdot 24 x^{6} + 7 (- 3 x) x^{6})$

Step 1.4.3

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7}) - 3 (7 \cdot 24 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 1.4.4

Combine terms.

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Step 1.4.4.1

Multiply $- 3$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 3 (7 \cdot 24 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 1.4.4.2

Multiply $7$ by $24$ .

$72 x^{2} + 9 x^{7} - 3 (168 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 1.4.4.3

Multiply $168$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (7 (- 3 x) x^{6})$

Step 1.4.4.4

Multiply $- 3$ by $7$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x \cdot x^{6})$

Step 1.4.4.5

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 1.4.4.5.1

Move $x^{6}$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 (x^{6} x))$

Step 1.4.4.5.2

Multiply $x^{6}$ by $x$ .

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Step 1.4.4.5.2.1

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

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 (x^{6} x^{1}))$

Step 1.4.4.5.2.2

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

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x^{6 + 1})$

Step 1.4.4.5.3

Add $6$ and $1$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x^{7})$

Step 1.4.4.6

Multiply $- 21$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} + 63 x^{7}$

Step 1.4.4.7

Add $9 x^{7}$ and $63 x^{7}$ .

$72 x^{2} + 72 x^{7} - 504 x^{6}$

Step 1.4.5

Reorder terms.

$72 x^{7} - 504 x^{6} + 72 x^{2}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $72 x^{7} - 504 x^{6} + 72 x^{2}$ with respect to $x$ is $\frac{d}{d x} [72 x^{7}] + \frac{d}{d x} [- 504 x^{6}] + \frac{d}{d x} [72 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (72 x^{7}) + \frac{d}{d x} (- 504 x^{6}) + \frac{d}{d x} (72 x^{2})$

Step 2.2

Evaluate $\frac{d}{d x} [72 x^{7}]$ .

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Step 2.2.1

Since $72$ is constant with respect to $x$ , the derivative of $72 x^{7}$ with respect to $x$ is $72 \frac{d}{d x} [x^{7}]$ .

$f'' (x) = 72 \frac{d}{d x} (x^{7}) + \frac{d}{d x} (- 504 x^{6}) + \frac{d}{d x} (72 x^{2})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f'' (x) = 72 (7 x^{6}) + \frac{d}{d x} (- 504 x^{6}) + \frac{d}{d x} (72 x^{2})$

Step 2.2.3

Multiply $7$ by $72$ .

$f'' (x) = 504 x^{6} + \frac{d}{d x} (- 504 x^{6}) + \frac{d}{d x} (72 x^{2})$

Step 2.3

Evaluate $\frac{d}{d x} [- 504 x^{6}]$ .

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Step 2.3.1

Since $- 504$ is constant with respect to $x$ , the derivative of $- 504 x^{6}$ with respect to $x$ is $- 504 \frac{d}{d x} [x^{6}]$ .

$f'' (x) = 504 x^{6} - 504 \frac{d}{d x} x^{6} + \frac{d}{d x} (72 x^{2})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = 504 x^{6} - 504 (6 x^{5}) + \frac{d}{d x} (72 x^{2})$

Step 2.3.3

Multiply $6$ by $- 504$ .

$f'' (x) = 504 x^{6} - 3024 x^{5} + \frac{d}{d x} (72 x^{2})$

Step 2.4

Evaluate $\frac{d}{d x} [72 x^{2}]$ .

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Step 2.4.1

Since $72$ is constant with respect to $x$ , the derivative of $72 x^{2}$ with respect to $x$ is $72 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 504 x^{6} - 3024 x^{5} + 72 \frac{d}{d x} (x^{2})$

Step 2.4.2

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

$f'' (x) = 504 x^{6} - 3024 x^{5} + 72 (2 x)$

Step 2.4.3

Multiply $2$ by $72$ .

$f'' (x) = 504 x^{6} - 3024 x^{5} + 144 x$

Step 3

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

$72 x^{7} - 504 x^{6} + 72 x^{2} = 0$

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

By the Sum Rule, the derivative of $24 x^{3} - 3 x^{4} \cdot (x^{3} (24 - 3 x))$ with respect to $x$ is $\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$ .

$\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 4.1.2

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

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Step 4.1.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$24 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 4.1.2.2

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

$24 (3 x^{2}) + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 4.1.2.3

Multiply $3$ by $24$ .

$72 x^{2} + \frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 3 x^{4} \cdot (x^{3} (24 - 3 x))]$ .

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Step 4.1.3.1

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

$72 x^{2} + \frac{d}{d x} [- 3 x^{3 + 4} \cdot (24 - 3 x)]$

Step 4.1.3.2

Add $3$ and $4$ .

$72 x^{2} + \frac{d}{d x} [- 3 x^{7} \cdot (24 - 3 x)]$

Step 4.1.3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{7} (24 - 3 x)$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{7} (24 - 3 x)]$ .

$72 x^{2} - 3 \frac{d}{d x} [x^{7} (24 - 3 x)]$

Step 4.1.3.4

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 $f (x) = x^{7}$ and $g (x) = 24 - 3 x$ .

$72 x^{2} - 3 (x^{7} \frac{d}{d x} [24 - 3 x] + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.5

By the Sum Rule, the derivative of $24 - 3 x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- 3 x]$ .

$72 x^{2} - 3 (x^{7} (\frac{d}{d x} [24] + \frac{d}{d x} [- 3 x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.6

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$72 x^{2} - 3 (x^{7} (0 + \frac{d}{d x} [- 3 x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.7

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

$72 x^{2} - 3 (x^{7} (0 - 3 \frac{d}{d x} [x]) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.8

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

$72 x^{2} - 3 (x^{7} (0 - 3 \cdot 1) + (24 - 3 x) \frac{d}{d x} [x^{7}])$

Step 4.1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$72 x^{2} - 3 (x^{7} (0 - 3 \cdot 1) + (24 - 3 x) (7 x^{6}))$

Step 4.1.3.10

Multiply $- 3$ by $1$ .

$72 x^{2} - 3 (x^{7} (0 - 3) + (24 - 3 x) (7 x^{6}))$

Step 4.1.3.11

Subtract $3$ from $0$ .

$72 x^{2} - 3 (x^{7} \cdot - 3 + (24 - 3 x) (7 x^{6}))$

Step 4.1.3.12

Move $- 3$ to the left of $x^{7}$ .

$72 x^{2} - 3 (- 3 \cdot x^{7} + (24 - 3 x) (7 x^{6}))$

Step 4.1.3.13

Move $7$ to the left of $24 - 3 x$ .

$72 x^{2} - 3 (- 3 x^{7} + 7 (24 - 3 x) x^{6})$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7} + (7 \cdot 24 + 7 (- 3 x)) x^{6})$

Step 4.1.4.2

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7} + 7 \cdot 24 x^{6} + 7 (- 3 x) x^{6})$

Step 4.1.4.3

Apply the distributive property.

$72 x^{2} - 3 (- 3 x^{7}) - 3 (7 \cdot 24 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 4.1.4.4

Combine terms.

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Step 4.1.4.4.1

Multiply $- 3$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 3 (7 \cdot 24 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 4.1.4.4.2

Multiply $7$ by $24$ .

$72 x^{2} + 9 x^{7} - 3 (168 x^{6}) - 3 (7 (- 3 x) x^{6})$

Step 4.1.4.4.3

Multiply $168$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (7 (- 3 x) x^{6})$

Step 4.1.4.4.4

Multiply $- 3$ by $7$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x \cdot x^{6})$

Step 4.1.4.4.5

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 4.1.4.4.5.1

Move $x^{6}$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 (x^{6} x))$

Step 4.1.4.4.5.2

Multiply $x^{6}$ by $x$ .

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Step 4.1.4.4.5.2.1

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

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 (x^{6} x^{1}))$

Step 4.1.4.4.5.2.2

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

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x^{6 + 1})$

Step 4.1.4.4.5.3

Add $6$ and $1$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} - 3 (- 21 x^{7})$

Step 4.1.4.4.6

Multiply $- 21$ by $- 3$ .

$72 x^{2} + 9 x^{7} - 504 x^{6} + 63 x^{7}$

Step 4.1.4.4.7

Add $9 x^{7}$ and $63 x^{7}$ .

$72 x^{2} + 72 x^{7} - 504 x^{6}$

Step 4.1.4.5

Reorder terms.

$f' (x) = 72 x^{7} - 504 x^{6} + 72 x^{2}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $72 x^{7} - 504 x^{6} + 72 x^{2}$ .

$72 x^{7} - 504 x^{6} + 72 x^{2}$

Step 5

Set the first derivative equal to $0$ then solve the equation $72 x^{7} - 504 x^{6} + 72 x^{2} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$72 x^{7} - 504 x^{6} + 72 x^{2} = 0$

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.60223321, 0, 0.62944187, 6.9995834$

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.

$x = - 0.60223321, 0, 0.62944187, 6.9995834$

Step 8

Evaluate the second derivative at $x = - 0.60223321$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$504 {(- 0.60223321)}^{6} - 3024 {(- 0.60223321)}^{5} + 144 (- 0.60223321)$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raise $- 0.60223321$ to the power of $6$ .

$504 \cdot 0.04770767 - 3024 {(- 0.60223321)}^{5} + 144 (- 0.60223321)$

Step 9.1.2

Multiply $504$ by $0.04770767$ .

$24.04466629 - 3024 {(- 0.60223321)}^{5} + 144 (- 0.60223321)$

Step 9.1.3

Raise $- 0.60223321$ to the power of $5$ .

$24.04466629 - 3024 \cdot - 0.07921793 + 144 (- 0.60223321)$

Step 9.1.4

Multiply $- 3024$ by $- 0.07921793$ .

$24.04466629 + 239.55503398 + 144 (- 0.60223321)$

Step 9.1.5

Multiply $144$ by $- 0.60223321$ .

$24.04466629 + 239.55503398 - 86.72158264$

Step 9.2

Simplify by adding and subtracting.

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Step 9.2.1

Add $24.04466629$ and $239.55503398$ .

$263.59970027 - 86.72158264$

Step 9.2.2

Subtract $86.72158264$ from $263.59970027$ .

$176.87811762$

Step 10

$x = - 0.60223321$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 0.60223321$ is a local minimum

Step 11

Find the y-value when $x = - 0.60223321$ .

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Step 11.1

Replace the variable $x$ with $- 0.60223321$ in the expression.

$f (- 0.60223321) = 24 {(- 0.60223321)}^{3} - 3 {(- 0.60223321)}^{4} \cdot ({(- 0.60223321)}^{3} (24 - 3 \cdot - 0.60223321))$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raise $- 0.60223321$ to the power of $3$ .

$f (- 0.60223321) = 24 \cdot - 0.21842085 - 3 {(- 0.60223321)}^{4} \cdot ({(- 0.60223321)}^{3} (24 - 3 \cdot - 0.60223321))$

Step 11.2.1.2

Multiply $24$ by $- 0.21842085$ .

$f (- 0.60223321) = - 5.24210059 - 3 {(- 0.60223321)}^{4} \cdot ({(- 0.60223321)}^{3} (24 - 3 \cdot - 0.60223321))$

Step 11.2.1.3

Multiply ${(- 0.60223321)}^{4}$ by ${(- 0.60223321)}^{3}$ by adding the exponents.

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Step 11.2.1.3.1

Move ${(- 0.60223321)}^{3}$ .

$f (- 0.60223321) = - 5.24210059 - 3 ({(- 0.60223321)}^{3} {(- 0.60223321)}^{4}) \cdot (24 - 3 \cdot - 0.60223321)$

Step 11.2.1.3.2

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

$f (- 0.60223321) = - 5.24210059 - 3 {(- 0.60223321)}^{3 + 4} \cdot (24 - 3 \cdot - 0.60223321)$

Step 11.2.1.3.3

Add $3$ and $4$ .

$f (- 0.60223321) = - 5.24210059 - 3 {(- 0.60223321)}^{7} \cdot (24 - 3 \cdot - 0.60223321)$

Step 11.2.1.4

Raise $- 0.60223321$ to the power of $7$ .

$f (- 0.60223321) = - 5.24210059 - 3 \cdot - 0.02873114 \cdot (24 - 3 \cdot - 0.60223321)$

Step 11.2.1.5

Multiply $- 3$ by $- 0.02873114$ .

$f (- 0.60223321) = - 5.24210059 + 0.08619343 \cdot (24 - 3 \cdot - 0.60223321)$

Step 11.2.1.6

Multiply $- 3$ by $- 0.60223321$ .

$f (- 0.60223321) = - 5.24210059 + 0.08619343 \cdot (24 + 1.80669963)$

Step 11.2.1.7

Add $24$ and $1.80669963$ .

$f (- 0.60223321) = - 5.24210059 + 0.08619343 \cdot 25.80669963$

Step 11.2.1.8

Multiply $0.08619343$ by $25.80669963$ .

$f (- 0.60223321) = - 5.24210059 + 2.22436801$

Step 11.2.2

Add $- 5.24210059$ and $2.22436801$ .

$f (- 0.60223321) = - 3.01773257$

Step 11.2.3

The final answer is $- 3.01773257$ .

$y = - 3.01773257$

Step 12

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.

$504 {(0)}^{6} - 3024 {(0)}^{5} + 144 (0)$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

$504 \cdot 0 - 3024 {(0)}^{5} + 144 (0)$

Step 13.1.2

Multiply $504$ by $0$ .

$0 - 3024 {(0)}^{5} + 144 (0)$

Step 13.1.3

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

$0 - 3024 \cdot 0 + 144 (0)$

Step 13.1.4

Multiply $- 3024$ by $0$ .

$0 + 0 + 144 (0)$

Step 13.1.5

Multiply $144$ by $0$ .

$0 + 0 + 0$

Step 13.2

Simplify by adding numbers.

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Step 13.2.1

Add $0$ and $0$ .

$0 + 0$

Step 13.2.2

Add $0$ and $0$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 0.60223321) \cup (- 0.60223321, 0) \cup (0, 0.62944187) \cup (0.62944187, 6.9995834) \cup (6.9995834, \infty)$

Step 14.2

Substitute any number, such as $- 3$ , from the interval $(- \infty, - 0.60223321)$ in the first derivative $72 x^{7} - 504 x^{6} + 72 x^{2}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = 72 {(- 3)}^{7} - 504 {(- 3)}^{6} + 72 {(- 3)}^{2}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify each term.

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Step 14.2.2.1.1

Raise $- 3$ to the power of $7$ .

$f' (- 3) = 72 \cdot - 2187 - 504 {(- 3)}^{6} + 72 {(- 3)}^{2}$

Step 14.2.2.1.2

Multiply $72$ by $- 2187$ .

$f' (- 3) = - 157464 - 504 {(- 3)}^{6} + 72 {(- 3)}^{2}$

Step 14.2.2.1.3

Raise $- 3$ to the power of $6$ .

$f' (- 3) = - 157464 - 504 \cdot 729 + 72 {(- 3)}^{2}$

Step 14.2.2.1.4

Multiply $- 504$ by $729$ .

$f' (- 3) = - 157464 - 367416 + 72 {(- 3)}^{2}$

Step 14.2.2.1.5

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

$f' (- 3) = - 157464 - 367416 + 72 \cdot 9$

Step 14.2.2.1.6

Multiply $72$ by $9$ .

$f' (- 3) = - 157464 - 367416 + 648$

Step 14.2.2.2

Simplify by adding and subtracting.

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Step 14.2.2.2.1

Subtract $367416$ from $- 157464$ .

$f' (- 3) = - 524880 + 648$

Step 14.2.2.2.2

Add $- 524880$ and $648$ .

$f' (- 3) = - 524232$

Step 14.2.2.3

The final answer is $- 524232$ .

$- 524232$

Step 14.3

Substitute any number, such as $- 0.3$ , from the interval $(- 0.60223321, 0)$ in the first derivative $72 x^{7} - 504 x^{6} + 72 x^{2}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $- 0.3$ in the expression.

$f' (- 0.3) = 72 {(- 0.3)}^{7} - 504 {(- 0.3)}^{6} + 72 {(- 0.3)}^{2}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify each term.

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Step 14.3.2.1.1

Raise $- 0.3$ to the power of $7$ .

$f' (- 0.3) = 72 \cdot - 0.0002187 - 504 {(- 0.3)}^{6} + 72 {(- 0.3)}^{2}$

Step 14.3.2.1.2

Multiply $72$ by $- 0.0002187$ .

$f' (- 0.3) = - 0.0157464 - 504 {(- 0.3)}^{6} + 72 {(- 0.3)}^{2}$

Step 14.3.2.1.3

Raise $- 0.3$ to the power of $6$ .

$f' (- 0.3) = - 0.0157464 - 504 \cdot 0.000729 + 72 {(- 0.3)}^{2}$

Step 14.3.2.1.4

Multiply $- 504$ by $0.000729$ .

$f' (- 0.3) = - 0.0157464 - 0.367416 + 72 {(- 0.3)}^{2}$

Step 14.3.2.1.5

Raise $- 0.3$ to the power of $2$ .

$f' (- 0.3) = - 0.0157464 - 0.367416 + 72 \cdot 0.09$

Step 14.3.2.1.6

Multiply $72$ by $0.09$ .

$f' (- 0.3) = - 0.0157464 - 0.367416 + 6.48$

Step 14.3.2.2

Simplify by adding and subtracting.

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Step 14.3.2.2.1

Subtract $0.367416$ from $- 0.0157464$ .

$f' (- 0.3) = - 0.3831624 + 6.48$

Step 14.3.2.2.2

Add $- 0.3831624$ and $6.48$ .

$f' (- 0.3) = 6.0968376$

Step 14.3.2.3

The final answer is $6.0968376$ .

$6.0968376$

Step 14.4

Substitute any number, such as $0.31$ , from the interval $(0, 0.62944187)$ in the first derivative $72 x^{7} - 504 x^{6} + 72 x^{2}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $0.31$ in the expression.

$f' (0.31) = 72 {(0.31)}^{7} - 504 {(0.31)}^{6} + 72 {(0.31)}^{2}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify each term.

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Step 14.4.2.1.1

Raise $0.31$ to the power of $7$ .

$f' (0.31) = 72 \cdot 0.00027512 - 504 {(0.31)}^{6} + 72 {(0.31)}^{2}$

Step 14.4.2.1.2

Multiply $72$ by $0.00027512$ .

$f' (0.31) = 0.01980908 - 504 {(0.31)}^{6} + 72 {(0.31)}^{2}$

Step 14.4.2.1.3

Raise $0.31$ to the power of $6$ .

$f' (0.31) = 0.01980908 - 504 \cdot 0.0008875 + 72 {(0.31)}^{2}$

Step 14.4.2.1.4

Multiply $- 504$ by $0.0008875$ .

$f' (0.31) = 0.01980908 - 0.44730185 + 72 {(0.31)}^{2}$

Step 14.4.2.1.5

Raise $0.31$ to the power of $2$ .

$f' (0.31) = 0.01980908 - 0.44730185 + 72 \cdot 0.0961$

Step 14.4.2.1.6

Multiply $72$ by $0.0961$ .

$f' (0.31) = 0.01980908 - 0.44730185 + 6.9192$

Step 14.4.2.2

Simplify by adding and subtracting.

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Step 14.4.2.2.1

Subtract $0.44730185$ from $0.01980908$ .

$f' (0.31) = - 0.42749277 + 6.9192$

Step 14.4.2.2.2

Add $- 0.42749277$ and $6.9192$ .

$f' (0.31) = 6.49170722$

Step 14.4.2.3

The final answer is $6.49170722$ .

$6.49170722$

Step 14.5

Substitute any number, such as $4$ , from the interval $(0.62944187, 6.9995834)$ in the first derivative $72 x^{7} - 504 x^{6} + 72 x^{2}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 72 {(4)}^{7} - 504 {(4)}^{6} + 72 {(4)}^{2}$

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Simplify each term.

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Step 14.5.2.1.1

Raise $4$ to the power of $7$ .

$f' (4) = 72 \cdot 16384 - 504 {(4)}^{6} + 72 {(4)}^{2}$

Step 14.5.2.1.2

Multiply $72$ by $16384$ .

$f' (4) = 1179648 - 504 {(4)}^{6} + 72 {(4)}^{2}$

Step 14.5.2.1.3

Raise $4$ to the power of $6$ .

$f' (4) = 1179648 - 504 \cdot 4096 + 72 {(4)}^{2}$

Step 14.5.2.1.4

Multiply $- 504$ by $4096$ .

$f' (4) = 1179648 - 2064384 + 72 {(4)}^{2}$

Step 14.5.2.1.5

Raise $4$ to the power of $2$ .

$f' (4) = 1179648 - 2064384 + 72 \cdot 16$

Step 14.5.2.1.6

Multiply $72$ by $16$ .

$f' (4) = 1179648 - 2064384 + 1152$

Step 14.5.2.2

Simplify by adding and subtracting.

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Step 14.5.2.2.1

Subtract $2064384$ from $1179648$ .

$f' (4) = - 884736 + 1152$

Step 14.5.2.2.2

Add $- 884736$ and $1152$ .

$f' (4) = - 883584$

Step 14.5.2.3

The final answer is $- 883584$ .

$- 883584$

Step 14.6

Substitute any number, such as $9$ , from the interval $(6.9995834, \infty)$ in the first derivative $72 x^{7} - 504 x^{6} + 72 x^{2}$ to check if the result is negative or positive.

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Step 14.6.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = 72 {(9)}^{7} - 504 {(9)}^{6} + 72 {(9)}^{2}$

Step 14.6.2

Simplify the result.

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Step 14.6.2.1

Simplify each term.

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Step 14.6.2.1.1

Raise $9$ to the power of $7$ .

$f' (9) = 72 \cdot 4782969 - 504 {(9)}^{6} + 72 {(9)}^{2}$

Step 14.6.2.1.2

Multiply $72$ by $4782969$ .

$f' (9) = 344373768 - 504 {(9)}^{6} + 72 {(9)}^{2}$

Step 14.6.2.1.3

Raise $9$ to the power of $6$ .

$f' (9) = 344373768 - 504 \cdot 531441 + 72 {(9)}^{2}$

Step 14.6.2.1.4

Multiply $- 504$ by $531441$ .

$f' (9) = 344373768 - 267846264 + 72 {(9)}^{2}$

Step 14.6.2.1.5

Raise $9$ to the power of $2$ .

$f' (9) = 344373768 - 267846264 + 72 \cdot 81$

Step 14.6.2.1.6

Multiply $72$ by $81$ .

$f' (9) = 344373768 - 267846264 + 5832$

Step 14.6.2.2

Simplify by adding and subtracting.

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Step 14.6.2.2.1

Subtract $267846264$ from $344373768$ .

$f' (9) = 76527504 + 5832$

Step 14.6.2.2.2

Add $76527504$ and $5832$ .

$f' (9) = 76533336$

Step 14.6.2.3

The final answer is $76533336$ .

$76533336$

Step 14.7

Since the first derivative changed signs from negative to positive around $x = - 0.60223321$ , then $x = - 0.60223321$ is a local minimum.

$x = - 0.60223321$ is a local minimum

Step 14.8

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.9

Since the first derivative changed signs from positive to negative around $x = 0.62944187$ , then $x = 0.62944187$ is a local maximum.

$x = 0.62944187$ is a local maximum

Step 14.10

Since the first derivative changed signs from negative to positive around $x = 6.9995834$ , then $x = 6.9995834$ is a local minimum.

$x = 6.9995834$ is a local minimum

Step 14.11

These are the local extrema for $f (x) = 24 x^{3} - 3 x^{4} \cdot (x^{3} (24 - 3 x))$ .

$x = - 0.60223321$ is a local minimum

$x = 0.62944187$ is a local maximum

$x = 6.9995834$ is a local minimum

$x = - 0.60223321$ is a local minimum

$x = 0.62944187$ is a local maximum

$x = 6.9995834$ is a local minimum

Step 15