Calculus Examples

$h (x) = \sqrt[3]{x}$ ,

$- 1 \leq x \leq 8$

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

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x}$ as

$x^{\frac{1}{3}}$ .

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

Step 1.1.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = \frac{1}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1}$

Step 1.1.1.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 1.1.1.4

Combine

$- 1$ and

$\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 1.1.1.5

Combine the numerators over the common denominator.

$\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}$

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply

$- 1$ by

$3$ .

$\frac{1}{3} x^{\frac{1 - 3}{3}}$

Step 1.1.1.6.2

Subtract

$3$ from

$1$ .

$\frac{1}{3} x^{\frac{- 2}{3}}$

Step 1.1.1.7

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}}$

Step 1.1.1.8

Simplify.

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Step 1.1.1.8.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}}$

Step 1.1.1.8.2

Multiply

$\frac{1}{3}$ by

$\frac{1}{x^{\frac{2}{3}}}$ .

$f' (x) = \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.2

The first derivative of

$h (x)$ with respect to

$x$ is

$\frac{1}{3 x^{\frac{2}{3}}}$ .

$\frac{1}{3 x^{\frac{2}{3}}}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{1}{3 x^{\frac{2}{3}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{1}{3 x^{\frac{2}{3}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since

$1 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Apply the rule

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{3 \sqrt[3]{x^{2}}}$

Step 1.3.2

Set the denominator in

$\frac{1}{3 \sqrt[3]{x^{2}}}$ equal to

$0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2}} = 0$

Step 1.3.3

Solve for

$x$ .

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Step 1.3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(3 \sqrt[3]{x^{2}})}^{3} = 0^{3}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x^{2}}$ as

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

${(3 x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

${(3 x^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to

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

$3^{3} {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.2

Raise

$3$ to the power of

$3$ .

$27 {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

${(x^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of

$3$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{2} = 0^{3}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$27 x^{2} = 0$

Step 1.3.3.3

Solve for

$x$ .

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Step 1.3.3.3.1

Divide each term in

$27 x^{2} = 0$ by

$27$ and simplify.

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Step 1.3.3.3.1.1

Divide each term in

$27 x^{2} = 0$ by

$27$ .

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2

Simplify the left side.

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Step 1.3.3.3.1.2.1

Cancel the common factor of

$27$ .

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Step 1.3.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{0}{27}$

Step 1.3.3.3.1.3

Simplify the right side.

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Step 1.3.3.3.1.3.1

Divide

$0$ by

$27$ .

$x^{2} = 0$

Step 1.3.3.3.2

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

$x = \pm \sqrt{0}$

Step 1.3.3.3.3

Simplify

$\pm \sqrt{0}$ .

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Step 1.3.3.3.3.1

Rewrite

$0$ as

$0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.3.3.3.3.2

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

$x = \pm 0$

Step 1.3.3.3.3.3

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 1.4

Evaluate

$\sqrt[3]{x}$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 0$ .

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Step 1.4.1.1

Substitute

$0$ for

$x$ .

$\sqrt[3]{0}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$\sqrt[3]{0}$

Step 1.4.1.2.2

Rewrite

$0$ as

$0^{3}$ .

$\sqrt[3]{0^{3}}$

Step 1.4.1.2.3

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

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at

$x = - 1$ .

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Step 2.1.1

Substitute

$- 1$ for

$x$ .

$\sqrt[3]{- 1}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Remove parentheses.

$\sqrt[3]{- 1}$

Step 2.1.2.2

Rewrite

$- 1$ as

${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3}}$

Step 2.1.2.3

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

$- 1$

Step 2.2

Evaluate at

$x = 8$ .

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Step 2.2.1

Substitute

$8$ for

$x$ .

$\sqrt[3]{8}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Remove parentheses.

$\sqrt[3]{8}$

Step 2.2.2.2

Rewrite

$8$ as

$2^{3}$ .

$\sqrt[3]{2^{3}}$

Step 2.2.2.3

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

$2$

Step 2.3

List all of the points.

$(- 1, - 1), (8, 2)$

Step 3

Compare the

$h (x)$ 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

$h (x)$ value and the minimum will occur at the lowest

$h (x)$ value.

Absolute Maximum:

$(8, 2)$

Absolute Minimum:

$(- 1, - 1)$

Step 4