Calculus Examples

$f (x) = \sqrt[3]{x - 8} - 1$ ;

$[0, 7]$

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

$\sqrt[3]{x - 8} - 1$ with respect to

$x$ is

$\frac{d}{d x} [\sqrt[3]{x - 8}] + \frac{d}{d x} [- 1]$ .

$f' (x) = \frac{d}{d x} (\sqrt[3]{x - 8}) + \frac{d}{d x} (- 1)$

Step 1.1.1.2

Evaluate

$\frac{d}{d x} [\sqrt[3]{x - 8}]$ .

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Step 1.1.1.2.1

Use

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

$\sqrt[3]{x - 8}$ as

${(x - 8)}^{\frac{1}{3}}$ .

$f' (x) = \frac{d}{d x} ({(x - 8)}^{\frac{1}{3}}) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.2

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

$g (x) = x - 8$ .

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Step 1.1.1.2.2.1

To apply the Chain Rule, set

$u$ as

$x - 8$ .

$f' (x) = \frac{d}{d u} (u^{\frac{1}{3}}) \frac{d}{d x} (x - 8) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.2.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

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

$f' (x) = \frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} (x - 8) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.2.3

Replace all occurrences of

$u$ with

$x - 8$ .

$f' (x) = \frac{1}{3} \cdot {(x - 8)}^{\frac{1}{3} - 1} \frac{d}{d x} (x - 8) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.3

By the Sum Rule, the derivative of

$x - 8$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1}{3} - 1} (\frac{d}{d x} (x) + \frac{d}{d x} (- 8))) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1}{3} - 1} (1 + \frac{d}{d x} (- 8))) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.5

Since

$- 8$ is constant with respect to

$x$ , the derivative of

$- 8$ with respect to

$x$ is

$0$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1}{3} - 1} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.6

To write

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

$\frac{3}{3}$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.7

Combine

$- 1$ and

$\frac{3}{3}$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.8

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1 - 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.9

Simplify the numerator.

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Step 1.1.1.2.9.1

Multiply

$- 1$ by

$3$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{1 - 3}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.9.2

Subtract

$3$ from

$1$ .

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{\frac{- 2}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.10

Move the negative in front of the fraction.

$f' (x) = \frac{1}{3} \cdot ({(x - 8)}^{- \frac{2}{3}} (1 + 0)) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.11

Add

$1$ and

$0$ .

$f' (x) = \frac{1}{3} \cdot {(x - 8)}^{- \frac{2}{3}} \cdot 1 + \frac{d}{d x} (- 1)$

Step 1.1.1.2.12

Combine

$\frac{1}{3}$ and

${(x - 8)}^{- \frac{2}{3}}$ .

$f' (x) = \frac{{(x - 8)}^{- \frac{2}{3}}}{3} \cdot 1 + \frac{d}{d x} (- 1)$

Step 1.1.1.2.13

Multiply

$\frac{{(x - 8)}^{- \frac{2}{3}}}{3}$ by

$1$ .

$f' (x) = \frac{{(x - 8)}^{- \frac{2}{3}}}{3} + \frac{d}{d x} (- 1)$

Step 1.1.1.2.14

Move

${(x - 8)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule

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

$f' (x) = \frac{1}{3 {(x - 8)}^{\frac{2}{3}}} + \frac{d}{d x} (- 1)$

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

$f' (x) = \frac{1}{3 {(x - 8)}^{\frac{2}{3}}} + 0$

Step 1.1.1.3.2

Add

$\frac{1}{3 {(x - 8)}^{\frac{2}{3}}}$ and

$0$ .

$f' (x) = \frac{1}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{1}{3 {(x - 8)}^{\frac{2}{3}}}$ .

$\frac{1}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{1}{3 {(x - 8)}^{\frac{2}{3}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{1}{3 {(x - 8)}^{\frac{2}{3}}} = 0$

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.

$\frac{1}{3 \sqrt[3]{{(x - 8)}^{2}}}$

Step 1.3.2

Set the denominator in

$\frac{1}{3 \sqrt[3]{{(x - 8)}^{2}}}$ equal to

$0$ to find where the expression is undefined.

$3 \sqrt[3]{{(x - 8)}^{2}} = 0$

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.

${(3 \sqrt[3]{{(x - 8)}^{2}})}^{3} = 0^{3}$

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

$\sqrt[3]{{(x - 8)}^{2}}$ as

${(x - 8)}^{\frac{2}{3}}$ .

${(3 {(x - 8)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

${(3 {(x - 8)}^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to

$3 {(x - 8)}^{\frac{2}{3}}$ .

$3^{3} {({(x - 8)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.2

Raise

$3$ to the power of

$3$ .

$27 {({(x - 8)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

${({(x - 8)}^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

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

$27 {(x - 8)}^{\frac{2}{3} \cdot 3} = 0^{3}$

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.

$27 {(x - 8)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 8)}^{2} = 0^{3}$

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

$0$ .

$27 {(x - 8)}^{2} = 0$

Step 1.3.3.3

Solve for

$x$ .

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Step 1.3.3.3.1

Divide each term in

$27 {(x - 8)}^{2} = 0$ by

$27$ and simplify.

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Step 1.3.3.3.1.1

Divide each term in

$27 {(x - 8)}^{2} = 0$ by

$27$ .

$\frac{27 {(x - 8)}^{2}}{27} = \frac{0}{27}$

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.

$\frac{27 {(x - 8)}^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2.1.2

Divide

${(x - 8)}^{2}$ by

$1$ .

${(x - 8)}^{2} = \frac{0}{27}$

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

$27$ .

${(x - 8)}^{2} = 0$

Step 1.3.3.3.2

Set the

$x - 8$ equal to

$0$ .

$x - 8 = 0$

Step 1.3.3.3.3

Add

$8$ to both sides of the equation.

$x = 8$

Step 1.4

Evaluate

$\sqrt[3]{x - 8} - 1$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 8$ .

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Step 1.4.1.1

Substitute

$8$ for

$x$ .

$\sqrt[3]{(8) - 8} - 1$

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

Subtract

$8$ from

$8$ .

$\sqrt[3]{0} - 1$

Step 1.4.1.2.1.2

Rewrite

$0$ as

$0^{3}$ .

$\sqrt[3]{0^{3}} - 1$

Step 1.4.1.2.1.3

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

$0 - 1$

Step 1.4.1.2.2

Subtract

$1$ from

$0$ .

$- 1$

Step 1.4.2

List all of the points.

$(8, - 1)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at

$x = 0$ .

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Step 3.1.1

Substitute

$0$ for

$x$ .

$\sqrt[3]{(0) - 8} - 1$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Subtract

$8$ from

$0$ .

$\sqrt[3]{- 8} - 1$

Step 3.1.2.1.2

Rewrite

$- 8$ as

${(- 2)}^{3}$ .

$\sqrt[3]{{(- 2)}^{3}} - 1$

Step 3.1.2.1.3

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

$- 2 - 1$

Step 3.1.2.2

Subtract

$1$ from

$- 2$ .

$- 3$

Step 3.2

Evaluate at

$x = 7$ .

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Step 3.2.1

Substitute

$7$ for

$x$ .

$\sqrt[3]{(7) - 8} - 1$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Subtract

$8$ from

$7$ .

$\sqrt[3]{- 1} - 1$

Step 3.2.2.1.2

Rewrite

$- 1$ as

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

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

Step 3.2.2.1.3

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

$- 1 - 1$

Step 3.2.2.2

Subtract

$1$ from

$- 1$ .

$- 2$

Step 3.3

List all of the points.

$(0, - 3), (7, - 2)$

Step 4

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:

$(7, - 2)$

Absolute Minimum:

$(0, - 3)$

Step 5