Calculus Examples

$f (x) = \frac{1}{3} x^{3} - 4 x$ , $[- 8, 8]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $\frac{1}{3} x^{3} - 4 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 4 x]$ .

$\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.2

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

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.4

Combine $\frac{3}{3}$ and $x^{2}$ .

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.1.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.5.2

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

$x^{2} + \frac{d}{d x} [- 4 x]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$x^{2} - 4 \frac{d}{d x} [x]$

Step 1.1.1.3.2

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

$x^{2} - 4 \cdot 1$

Step 1.1.1.3.3

Multiply $- 4$ by $1$ .

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

Step 1.1.2

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

$x^{2} - 4$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$x^{2} - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 1.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 1.2.4

Simplify $\pm \sqrt{4}$ .

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Step 1.2.4.1

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

$x = \pm \sqrt{2^{2}}$

Step 1.2.4.2

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

$x = \pm 2$

Step 1.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 1.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 1.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $\frac{1}{3} x^{3} - 4 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 2$ .

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Step 1.4.1.1

Substitute $2$ for $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

$\frac{1}{3} \cdot 8 - 4 \cdot 2$

Step 1.4.1.2.1.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} - 4 \cdot 2$

Step 1.4.1.2.1.3

Multiply $- 4$ by $2$ .

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

Step 1.4.1.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - 8 \cdot \frac{3}{3}$

Step 1.4.1.2.3

Combine $- 8$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{- 8 \cdot 3}{3}$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{8 - 8 \cdot 3}{3}$

Step 1.4.1.2.5

Simplify the numerator.

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Step 1.4.1.2.5.1

Multiply $- 8$ by $3$ .

$\frac{8 - 24}{3}$

Step 1.4.1.2.5.2

Subtract $24$ from $8$ .

$\frac{- 16}{3}$

Step 1.4.1.2.6

Move the negative in front of the fraction.

$- \frac{16}{3}$

Step 1.4.2

Evaluate at $x = - 2$ .

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Step 1.4.2.1

Substitute $- 2$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

$\frac{1}{3} \cdot - 8 - 4 \cdot - 2$

Step 1.4.2.2.1.2

Combine $\frac{1}{3}$ and $- 8$ .

$\frac{- 8}{3} - 4 \cdot - 2$

Step 1.4.2.2.1.3

Move the negative in front of the fraction.

$- \frac{8}{3} - 4 \cdot - 2$

Step 1.4.2.2.1.4

Multiply $- 4$ by $- 2$ .

$- \frac{8}{3} + 8$

Step 1.4.2.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{8}{3} + 8 \cdot \frac{3}{3}$

Step 1.4.2.2.3

Combine $8$ and $\frac{3}{3}$ .

$- \frac{8}{3} + \frac{8 \cdot 3}{3}$

Step 1.4.2.2.4

Combine the numerators over the common denominator.

$\frac{- 8 + 8 \cdot 3}{3}$

Step 1.4.2.2.5

Simplify the numerator.

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Step 1.4.2.2.5.1

Multiply $8$ by $3$ .

$\frac{- 8 + 24}{3}$

Step 1.4.2.2.5.2

Add $- 8$ and $24$ .

$\frac{16}{3}$

Step 1.4.3

List all of the points.

$(2, - \frac{16}{3}), (- 2, \frac{16}{3})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 8$ .

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Step 2.1.1

Substitute $- 8$ for $x$ .

$\frac{1}{3} \cdot {(- 8)}^{3} - 4 \cdot - 8$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Raise $- 8$ to the power of $3$ .

$\frac{1}{3} \cdot - 512 - 4 \cdot - 8$

Step 2.1.2.1.2

Combine $\frac{1}{3}$ and $- 512$ .

$\frac{- 512}{3} - 4 \cdot - 8$

Step 2.1.2.1.3

Move the negative in front of the fraction.

$- \frac{512}{3} - 4 \cdot - 8$

Step 2.1.2.1.4

Multiply $- 4$ by $- 8$ .

$- \frac{512}{3} + 32$

Step 2.1.2.2

To write $32$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{512}{3} + 32 \cdot \frac{3}{3}$

Step 2.1.2.3

Combine $32$ and $\frac{3}{3}$ .

$- \frac{512}{3} + \frac{32 \cdot 3}{3}$

Step 2.1.2.4

Combine the numerators over the common denominator.

$\frac{- 512 + 32 \cdot 3}{3}$

Step 2.1.2.5

Simplify the numerator.

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Step 2.1.2.5.1

Multiply $32$ by $3$ .

$\frac{- 512 + 96}{3}$

Step 2.1.2.5.2

Add $- 512$ and $96$ .

$\frac{- 416}{3}$

Step 2.1.2.6

Move the negative in front of the fraction.

$- \frac{416}{3}$

Step 2.2

Evaluate at $x = 8$ .

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Step 2.2.1

Substitute $8$ for $x$ .

$\frac{1}{3} \cdot {(8)}^{3} - 4 \cdot 8$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Raise $8$ to the power of $3$ .

$\frac{1}{3} \cdot 512 - 4 \cdot 8$

Step 2.2.2.1.2

Combine $\frac{1}{3}$ and $512$ .

$\frac{512}{3} - 4 \cdot 8$

Step 2.2.2.1.3

Multiply $- 4$ by $8$ .

$\frac{512}{3} - 32$

Step 2.2.2.2

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{512}{3} - 32 \cdot \frac{3}{3}$

Step 2.2.2.3

Combine $- 32$ and $\frac{3}{3}$ .

$\frac{512}{3} + \frac{- 32 \cdot 3}{3}$

Step 2.2.2.4

Combine the numerators over the common denominator.

$\frac{512 - 32 \cdot 3}{3}$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $- 32$ by $3$ .

$\frac{512 - 96}{3}$

Step 2.2.2.5.2

Subtract $96$ from $512$ .

$\frac{416}{3}$

Step 2.3

List all of the points.

$(- 8, - \frac{416}{3}), (8, \frac{416}{3})$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(8, \frac{416}{3})$

Absolute Minimum: $(- 8, - \frac{416}{3})$

Step 4