Calculus Examples

f (x) = 2 \sqrt[3]{x - 1} + 3

$f (x) = 2 \sqrt[3]{x - 1} + 3$ ;

[- 7, 9]

$[- 7, 9]$

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

2 \sqrt[3]{x - 1} + 3

$2 \sqrt[3]{x - 1} + 3$ with respect to

x

$x$ is

\frac{d}{d x} [2 \sqrt[3]{x - 1}] + \frac{d}{d x} [3]

$\frac{d}{d x} [2 \sqrt[3]{x - 1}] + \frac{d}{d x} [3]$ .

f' (x) = \frac{d}{d x} (2 \sqrt[3]{x - 1}) + \frac{d}{d x} (3)

$f' (x) = \frac{d}{d x} (2 \sqrt[3]{x - 1}) + \frac{d}{d x} (3)$

Step 1.1.1.2

Evaluate

\frac{d}{d x} [2 \sqrt[3]{x - 1}]

$\frac{d}{d x} [2 \sqrt[3]{x - 1}]$ .

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Step 1.1.1.2.1

Use

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

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

\sqrt[3]{x - 1}

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

{(x - 1)}^{\frac{1}{3}}

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

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

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

Step 1.1.1.2.2

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

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

$2 {(x - 1)}^{\frac{1}{3}}$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.1.2.3

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

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

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

g (x) = x - 1

$g (x) = x - 1$ .

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Step 1.1.1.2.3.1

To apply the Chain Rule, set

u

$u$ as

x - 1

$x - 1$ .

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

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

Step 1.1.1.2.3.2

Differentiate using the Power Rule which states that

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

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

n u^{n - 1}

$n u^{n - 1}$ where

n = \frac{1}{3}

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

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

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

Step 1.1.1.2.3.3

Replace all occurrences of

u

$u$ with

x - 1

$x - 1$ .

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

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

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

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

Step 1.1.1.2.4

By the Sum Rule, the derivative of

x - 1

$x - 1$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 1]

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

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

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

Step 1.1.1.2.5

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.1.1.2.6

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.1.1.2.7

To write

- 1

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

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 1.1.1.2.8

Combine

- 1

$- 1$ and

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 1.1.1.2.9

Combine the numerators over the common denominator.

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

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

Step 1.1.1.2.10

Simplify the numerator.

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Step 1.1.1.2.10.1

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 1.1.1.2.10.2

Subtract

3

$3$ from

1

$1$ .

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

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

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

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

Step 1.1.1.2.11

Move the negative in front of the fraction.

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

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

Step 1.1.1.2.12

Add

1

$1$ and

0

$0$ .

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

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

Step 1.1.1.2.13

Combine

\frac{1}{3}

$\frac{1}{3}$ and

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

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

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

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

Step 1.1.1.2.14

Multiply

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

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

1

$1$ .

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

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

Step 1.1.1.2.15

Move

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

${(x - 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule

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

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

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

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

Step 1.1.1.2.16

Combine

2

$2$ and

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

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

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

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

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

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3

$3$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.1.1.3.2

Add

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

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

0

$0$ .

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

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

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

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

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

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

Step 1.1.2

The first derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

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

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

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

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

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

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

Step 1.2

Set the first derivative equal to

0

$0$ then solve the equation

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

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

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Step 1.2.1

Set the first derivative equal to

0

$0$ .

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

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

Step 1.2.2

Set the numerator equal to zero.

2 = 0

$2 = 0$

Step 1.2.3

Since

2 \neq 0

$2 \neq 0$ , 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}}

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

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

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

Step 1.3.2

Set the denominator in

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

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

0

$0$ to find where the expression is undefined.

3 \sqrt[3]{{(x - 1)}^{2}} = 0

$3 \sqrt[3]{{(x - 1)}^{2}} = 0$

Step 1.3.3

Solve for

x

$x$ .

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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 - 1)}^{2}})}^{3} = 0^{3}

${(3 \sqrt[3]{{(x - 1)}^{2}})}^{3} = 0^{3}$

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}}

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

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

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

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

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

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

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

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

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

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Step 1.3.3.2.2.1.1

Apply the product rule to

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

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

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

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

Step 1.3.3.2.2.1.2

Raise

3

$3$ to the power of

3

$3$ .

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

$27 {({(x - 1)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

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

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

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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}

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

27 {(x - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}

$27 {(x - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of

3

$3$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(x - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 1)}^{2} = 0^{3}$

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

$0$ .

$27 {(x - 1)}^{2} = 0$

Step 1.3.3.3

Solve for

$x$ .

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Step 1.3.3.3.1

Divide each term in

$27 {(x - 1)}^{2} = 0$ by

$27$ and simplify.

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Step 1.3.3.3.1.1

Divide each term in

$27 {(x - 1)}^{2} = 0$ by

$27$ .

$\frac{27 {(x - 1)}^{2}}{27} = \frac{0}{27}$

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.

$\frac{27 {(x - 1)}^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2.1.2

Divide

${(x - 1)}^{2}$ by

$1$ .

${(x - 1)}^{2} = \frac{0}{27}$

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

$27$ .

${(x - 1)}^{2} = 0$

Step 1.3.3.3.2

Set the

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 1.3.3.3.3

Add

$1$ to both sides of the equation.

$x = 1$

Step 1.4

Evaluate

$2 \sqrt[3]{x - 1} + 3$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 1$ .

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Step 1.4.1.1

Substitute

$1$ for

$x$ .

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

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

$1$ from

$1$ .

$2 \sqrt[3]{0} + 3$

Step 1.4.1.2.1.2

Rewrite

$0$ as

$0^{3}$ .

$2 \sqrt[3]{0^{3}} + 3$

Step 1.4.1.2.1.3

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

$2 \cdot 0 + 3$

Step 1.4.1.2.1.4

Multiply

$2$ by

$0$ .

$0 + 3$

Step 1.4.1.2.2

Add

$0$ and

$3$ .

$3$

Step 1.4.2

List all of the points.

$(1, 3)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at

$x = - 7$ .

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Step 2.1.1

Substitute

$- 7$ for

$x$ .

$2 \sqrt[3]{(- 7) - 1} + 3$

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

Subtract

$1$ from

$- 7$ .

$2 \sqrt[3]{- 8} + 3$

Step 2.1.2.1.2

Rewrite

$- 8$ as

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

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

Step 2.1.2.1.3

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

$2 \cdot - 2 + 3$

Step 2.1.2.1.4

Multiply

$2$ by

$- 2$ .

$- 4 + 3$

Step 2.1.2.2

Add

$- 4$ and

$3$ .

$- 1$

Step 2.2

Evaluate at

$x = 9$ .

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Step 2.2.1

Substitute

$9$ for

$x$ .

$2 \sqrt[3]{(9) - 1} + 3$

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

Subtract

$1$ from

$9$ .

$2 \sqrt[3]{8} + 3$

Step 2.2.2.1.2

Rewrite

$8$ as

$2^{3}$ .

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

Step 2.2.2.1.3

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

$2 \cdot 2 + 3$

Step 2.2.2.1.4

Multiply

$2$ by

$2$ .

$4 + 3$

Step 2.2.2.2

Add

$4$ and

$3$ .

$7$

Step 2.3

List all of the points.

$(- 7, - 1), (9, 7)$

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:

$(9, 7)$

Absolute Minimum:

$(- 7, - 1)$

Step 4