Calculus Examples

$f (x) = 12 x^{\frac{2}{3}} + 8$ , $[- 8, 1]$

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 $12 x^{\frac{2}{3}} + 8$ with respect to $x$ is $\frac{d}{d x} [12 x^{\frac{2}{3}}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [12 x^{\frac{2}{3}}] + \frac{d}{d x} [8]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [12 x^{\frac{2}{3}}]$ .

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Step 1.1.1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{\frac{2}{3}}$ with respect to $x$ is $12 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$12 \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [8]$

Step 1.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$12 (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [8]$

Step 1.1.1.2.3

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

$12 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.4

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

$12 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.5

Combine the numerators over the common denominator.

$12 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.6

Simplify the numerator.

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Step 1.1.1.2.6.1

Multiply $- 1$ by $3$ .

$12 (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.6.2

Subtract $3$ from $2$ .

$12 (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.7

Move the negative in front of the fraction.

$12 (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [8]$

Step 1.1.1.2.8

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

$12 \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [8]$

Step 1.1.1.2.9

Combine $12$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{12 (2 x^{- \frac{1}{3}})}{3} + \frac{d}{d x} [8]$

Step 1.1.1.2.10

Multiply $2$ by $12$ .

$\frac{24 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [8]$

Step 1.1.1.2.11

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

$\frac{24}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [8]$

Step 1.1.1.2.12

Factor $3$ out of $24$ .

$\frac{3 \cdot 8}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [8]$

Step 1.1.1.2.13

Cancel the common factors.

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Step 1.1.1.2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$\frac{3 \cdot 8}{3 (x^{\frac{1}{3}})} + \frac{d}{d x} [8]$

Step 1.1.1.2.13.2

Cancel the common factor.

$\frac{3 \cdot 8}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [8]$

Step 1.1.1.2.13.3

Rewrite the expression.

$\frac{8}{x^{\frac{1}{3}}} + \frac{d}{d x} [8]$

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{8}{x^{\frac{1}{3}}} + 0$

Step 1.1.1.3.2

Add $\frac{8}{x^{\frac{1}{3}}}$ and $0$ .

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8}{x^{\frac{1}{3}}}$ .

$\frac{8}{x^{\frac{1}{3}}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{8}{x^{\frac{1}{3}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{8}{x^{\frac{1}{3}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$8 = 0$

Step 1.2.3

Since $8 \neq 0$ , 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

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

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

$\frac{8}{\sqrt[3]{x^{1}}}$

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{8}{\sqrt[3]{x}}$

Step 1.3.2

Set the denominator in $\frac{8}{\sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x} = 0$

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.

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

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

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

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Step 1.3.3.2.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 1.3.3.2.2.1.2

Simplify.

$x = 0^{3}$

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 $0$ .

$x = 0$

Step 1.4

Evaluate $12 x^{\frac{2}{3}} + 8$ 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 $x$ .

$12 {(0)}^{\frac{2}{3}} + 8$

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

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

$12 {(0^{3})}^{\frac{2}{3}} + 8$

Step 1.4.1.2.1.2

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

$12 \cdot 0^{3 (\frac{2}{3})} + 8$

Step 1.4.1.2.1.3

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.3.1

Cancel the common factor.

$12 \cdot 0^{3 (\frac{2}{3})} + 8$

Step 1.4.1.2.1.3.2

Rewrite the expression.

$12 \cdot 0^{2} + 8$

Step 1.4.1.2.1.4

Raising $0$ to any positive power yields $0$ .

$12 \cdot 0 + 8$

Step 1.4.1.2.1.5

Multiply $12$ by $0$ .

$0 + 8$

Step 1.4.1.2.2

Add $0$ and $8$ .

$8$

Step 1.4.2

List all of the points.

$(0, 8)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 8$ .

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Step 2.1.1

Substitute $- 8$ for $x$ .

$12 {(- 8)}^{\frac{2}{3}} + 8$

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

Rewrite $- 8$ as ${(- 2)}^{3}$ .

$12 {({(- 2)}^{3})}^{\frac{2}{3}} + 8$

Step 2.1.2.1.2

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

$12 {(- 2)}^{3 (\frac{2}{3})} + 8$

Step 2.1.2.1.3

Cancel the common factor of $3$ .

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Step 2.1.2.1.3.1

Cancel the common factor.

$12 {(- 2)}^{3 (\frac{2}{3})} + 8$

Step 2.1.2.1.3.2

Rewrite the expression.

$12 {(- 2)}^{2} + 8$

Step 2.1.2.1.4

Raise $- 2$ to the power of $2$ .

$12 \cdot 4 + 8$

Step 2.1.2.1.5

Multiply $12$ by $4$ .

$48 + 8$

Step 2.1.2.2

Add $48$ and $8$ .

$56$

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

$12 {(1)}^{\frac{2}{3}} + 8$

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

One to any power is one.

$12 \cdot 1 + 8$

Step 2.2.2.1.2

Multiply $12$ by $1$ .

$12 + 8$

Step 2.2.2.2

Add $12$ and $8$ .

$20$

Step 2.3

List all of the points.

$(- 8, 56), (1, 20)$

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: $(- 8, 56)$

Absolute Minimum: $(0, 8)$

Step 4