Calculus Examples

$f (x) = \sqrt[3]{x + 4}$ ; $[- 4, 4]$

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 $\sqrt[3]{x + 4}$ as ${(x + 4)}^{\frac{1}{3}}$ .

$f' (x) = \frac{d}{d x} ({(x + 4)}^{\frac{1}{3}})$

Step 1.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x + 4$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $x + 4$ .

$f' (x) = \frac{d}{d u} (u^{\frac{1}{3}}) \frac{d}{d x} (x + 4)$

Step 1.1.1.2.2

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

$f' (x) = \frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} (x + 4)$

Step 1.1.1.2.3

Replace all occurrences of $u$ with $x + 4$ .

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{1}{3} - 1} \frac{d}{d x} (x + 4)$

Step 1.1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.4

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

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.5

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{1 - 3}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.6.2

Subtract $3$ from $1$ .

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{\frac{- 2}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.7

Combine fractions.

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Step 1.1.1.7.1

Move the negative in front of the fraction.

$f' (x) = \frac{1}{3} \cdot {(x + 4)}^{- \frac{2}{3}} \frac{d}{d x} (x + 4)$

Step 1.1.1.7.2

Combine $\frac{1}{3}$ and ${(x + 4)}^{- \frac{2}{3}}$ .

$f' (x) = \frac{{(x + 4)}^{- \frac{2}{3}}}{3} \cdot \frac{d}{d x} (x + 4)$

Step 1.1.1.7.3

Move ${(x + 4)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}} \cdot \frac{d}{d x} (x + 4)$

Step 1.1.1.8

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

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (4))$

Step 1.1.1.9

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

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}} \cdot (1 + \frac{d}{d x} (4))$

Step 1.1.1.10

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

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}} \cdot (1 + 0)$

Step 1.1.1.11

Simplify the expression.

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Step 1.1.1.11.1

Add $1$ and $0$ .

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}} \cdot 1$

Step 1.1.1.11.2

Multiply $\frac{1}{3 {(x + 4)}^{\frac{2}{3}}}$ by $1$ .

$f' (x) = \frac{1}{3 {(x + 4)}^{\frac{2}{3}}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{3 {(x + 4)}^{\frac{2}{3}}}$ .

$\frac{1}{3 {(x + 4)}^{\frac{2}{3}}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{3 {(x + 4)}^{\frac{2}{3}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{1}{3 {(x + 4)}^{\frac{2}{3}}} = 0$

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.

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

Step 1.3.2

Set the denominator in $\frac{1}{3 \sqrt[3]{{(x + 4)}^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{{(x + 4)}^{2}} = 0$

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.

${(3 \sqrt[3]{{(x + 4)}^{2}})}^{3} = 0^{3}$

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 $\sqrt[3]{{(x + 4)}^{2}}$ as ${(x + 4)}^{\frac{2}{3}}$ .

${(3 {(x + 4)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify ${(3 {(x + 4)}^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to $3 {(x + 4)}^{\frac{2}{3}}$ .

$3^{3} {({(x + 4)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.2

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

$27 {({(x + 4)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in ${({(x + 4)}^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 {(x + 4)}^{\frac{2}{3} \cdot 3} = 0^{3}$

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.

$27 {(x + 4)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x + 4)}^{2} = 0^{3}$

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 $0$ .

$27 {(x + 4)}^{2} = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Divide each term in $27 {(x + 4)}^{2} = 0$ by $27$ and simplify.

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Step 1.3.3.3.1.1

Divide each term in $27 {(x + 4)}^{2} = 0$ by $27$ .

$\frac{27 {(x + 4)}^{2}}{27} = \frac{0}{27}$

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.

$\frac{27 {(x + 4)}^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2.1.2

Divide ${(x + 4)}^{2}$ by $1$ .

${(x + 4)}^{2} = \frac{0}{27}$

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 $27$ .

${(x + 4)}^{2} = 0$

Step 1.3.3.3.2

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.3.3.3.3

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.4

Evaluate $\sqrt[3]{x + 4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = - 4$ .

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Step 1.4.1.1

Substitute $- 4$ for $x$ .

$\sqrt[3]{(- 4) + 4}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Add $- 4$ and $4$ .

$\sqrt[3]{0}$

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.

$(- 4, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 4$ .

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Step 2.1.1

Substitute $- 4$ for $x$ .

$\sqrt[3]{(- 4) + 4}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Add $- 4$ and $4$ .

$\sqrt[3]{0}$

Step 2.1.2.2

Rewrite $0$ as $0^{3}$ .

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

Step 2.1.2.3

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

$0$

Step 2.2

Evaluate at $x = 4$ .

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Step 2.2.1

Substitute $4$ for $x$ .

$\sqrt[3]{(4) + 4}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Add $4$ and $4$ .

$\sqrt[3]{8}$

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.

$(- 4, 0), (4, 2)$

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: $(4, 2)$

Absolute Minimum: $(- 4, 0)$

Step 4