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