Calculus Examples

$f (x) = 2 \sqrt[3]{x - 1} + 3$ ; $[- 7, 9]$

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1}$ as ${(x - 1)}^{\frac{1}{3}}$ .

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

Step 1.1.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 {(x - 1)}^{\frac{1}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [{(x - 1)}^{\frac{1}{3}}]$ .

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

Step 1.1.1.2.3

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 $g (x) = x - 1$ .

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Step 1.1.1.2.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 1.1.1.2.3.2

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

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

Step 1.1.1.2.3.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 1.1.1.2.4

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.1.1.2.7

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

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

Step 1.1.1.2.8

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

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

Step 1.1.1.2.9

Combine the numerators over the common denominator.

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

Step 1.1.1.2.10

Simplify the numerator.

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Step 1.1.1.2.10.1

Multiply $- 1$ by $3$ .

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

Step 1.1.1.2.10.2

Subtract $3$ from $1$ .

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

Step 1.1.1.2.11

Move the negative in front of the fraction.

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

Step 1.1.1.2.12

Add $1$ and $0$ .

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

Step 1.1.1.2.13

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

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

Step 1.1.1.2.14

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

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

Step 1.1.1.2.15

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

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

Step 1.1.1.2.16

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

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$2 = 0$

Step 1.2.3

Since $2 \neq 0$ , 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 $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 1.3.2

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

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

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.

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

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(x - 1)}^{2}}$ as ${(x - 1)}^{\frac{2}{3}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

Apply the product rule to $3 {(x - 1)}^{\frac{2}{3}}$ .

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

Step 1.3.3.2.2.1.2

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

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

Step 1.3.3.2.2.1.3

Multiply the exponents in ${({(x - 1)}^{\frac{2}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

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

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

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.

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

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