Calculus Examples

$f (x) = 9 - 9 x - \frac{16}{x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $9 - 9 x - \frac{16}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 1.1.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 1.2

Evaluate $\frac{d}{d x} [- 9 x]$ .

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Step 1.2.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$0 - 9 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 1.2.2

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

$0 - 9 \cdot 1 + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 1.2.3

Multiply $- 9$ by $1$ .

$0 - 9 + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{16}{x^{2}}]$ .

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Step 1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16}{x^{2}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$0 - 9 - 16 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.3.2

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

$0 - 9 - 16 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$0 - 9 - 16 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.3.3.2

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

$0 - 9 - 16 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$0 - 9 - 16 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.3.4

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

$0 - 9 - 16 (- {(x^{2})}^{- 2} (2 x))$

Step 1.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 1.3.5.1

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

$0 - 9 - 16 (- x^{2 \cdot - 2} (2 x))$

Step 1.3.5.2

Multiply $2$ by $- 2$ .

$0 - 9 - 16 (- x^{- 4} (2 x))$

Step 1.3.6

Multiply $2$ by $- 1$ .

$0 - 9 - 16 (- 2 x^{- 4} x)$

Step 1.3.7

Raise $x$ to the power of $1$ .

$0 - 9 - 16 (- 2 (x^{1} x^{- 4}))$

Step 1.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - 9 - 16 (- 2 x^{1 - 4})$

Step 1.3.9

Subtract $4$ from $1$ .

$0 - 9 - 16 (- 2 x^{- 3})$

Step 1.3.10

Multiply $- 2$ by $- 16$ .

$0 - 9 + 32 x^{- 3}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - 9 + 32 \frac{1}{x^{3}}$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Subtract $9$ from $0$ .

$- 9 + 32 \frac{1}{x^{3}}$

Step 1.4.2.2

Combine $32$ and $\frac{1}{x^{3}}$ .

$- 9 + \frac{32}{x^{3}}$

Step 1.4.3

Reorder terms.

$\frac{32}{x^{3}} - 9$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{32}{x^{3}} - 9$ with respect to $x$ is $\frac{d}{d x} [\frac{32}{x^{3}}] + \frac{d}{d x} [- 9]$ .

$f'' (x) = \frac{d}{d x} (\frac{32}{x^{3}}) + \frac{d}{d x} (- 9)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{32}{x^{3}}]$ .

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Step 2.2.1

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{3}}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 32 \frac{d}{d x} (\frac{1}{x^{3}}) + \frac{d}{d x} (- 9)$

Step 2.2.2

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

$f'' (x) = 32 \frac{d}{d x} ({(x^{3})}^{- 1}) + \frac{d}{d x} (- 9)$

Step 2.2.3

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

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$f'' (x) = 32 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (- 9)$

Step 2.2.3.2

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

$f'' (x) = 32 (- u^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- 9)$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$f'' (x) = 32 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- 9)$

Step 2.2.4

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

$f'' (x) = 32 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (- 9)$

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = 32 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (- 9)$

Step 2.2.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 32 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (- 9)$

Step 2.2.6

Multiply $3$ by $- 1$ .

$f'' (x) = 32 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (- 9)$

Step 2.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 2.2.7.1

Move $x^{2}$ .

$f'' (x) = 32 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (- 9)$

Step 2.2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 32 (- 3 x^{2 - 6}) + \frac{d}{d x} (- 9)$

Step 2.2.7.3

Subtract $6$ from $2$ .

$f'' (x) = 32 (- 3 x^{- 4}) + \frac{d}{d x} (- 9)$

Step 2.2.8

Multiply $- 3$ by $32$ .

$f'' (x) = - 96 x^{- 4} + \frac{d}{d x} (- 9)$

Step 2.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$f'' (x) = - 96 x^{- 4} + 0$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 96 \frac{1}{x^{4}} + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 96$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{- 96}{x^{4}} + 0$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{96}{x^{4}} + 0$

Step 2.4.2.3

Add $- \frac{96}{x^{4}}$ and $0$ .

$f'' (x) = - \frac{96}{x^{4}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{32}{x^{3}} - 9 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $9 - 9 x - \frac{16}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 4.1.1.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 9 x]$ .

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Step 4.1.2.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$0 - 9 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 4.1.2.2

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

$0 - 9 \cdot 1 + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 4.1.2.3

Multiply $- 9$ by $1$ .

$0 - 9 + \frac{d}{d x} [- \frac{16}{x^{2}}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- \frac{16}{x^{2}}]$ .

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Step 4.1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16}{x^{2}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$0 - 9 - 16 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 4.1.3.2

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

$0 - 9 - 16 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 4.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 4.1.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$0 - 9 - 16 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 4.1.3.3.2

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

$0 - 9 - 16 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 4.1.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$0 - 9 - 16 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 4.1.3.4

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

$0 - 9 - 16 (- {(x^{2})}^{- 2} (2 x))$

Step 4.1.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 4.1.3.5.1

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

$0 - 9 - 16 (- x^{2 \cdot - 2} (2 x))$

Step 4.1.3.5.2

Multiply $2$ by $- 2$ .

$0 - 9 - 16 (- x^{- 4} (2 x))$

Step 4.1.3.6

Multiply $2$ by $- 1$ .

$0 - 9 - 16 (- 2 x^{- 4} x)$

Step 4.1.3.7

Raise $x$ to the power of $1$ .

$0 - 9 - 16 (- 2 (x^{1} x^{- 4}))$

Step 4.1.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - 9 - 16 (- 2 x^{1 - 4})$

Step 4.1.3.9

Subtract $4$ from $1$ .

$0 - 9 - 16 (- 2 x^{- 3})$

Step 4.1.3.10

Multiply $- 2$ by $- 16$ .

$0 - 9 + 32 x^{- 3}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - 9 + 32 \frac{1}{x^{3}}$

Step 4.1.4.2

Combine terms.

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Step 4.1.4.2.1

Subtract $9$ from $0$ .

$- 9 + 32 \frac{1}{x^{3}}$

Step 4.1.4.2.2

Combine $32$ and $\frac{1}{x^{3}}$ .

$- 9 + \frac{32}{x^{3}}$

Step 4.1.4.3

Reorder terms.

$f' (x) = \frac{32}{x^{3}} - 9$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{32}{x^{3}} - 9$ .

$\frac{32}{x^{3}} - 9$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{32}{x^{3}} - 9 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{32}{x^{3}} - 9 = 0$

Step 5.2

Add $9$ to both sides of the equation.

$\frac{32}{x^{3}} = 9$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{3}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 5.4

Multiply each term in $\frac{32}{x^{3}} = 9$ by $x^{3}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $\frac{32}{x^{3}} = 9$ by $x^{3}$ .

$\frac{32}{x^{3}} x^{3} = 9 x^{3}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{32}{x^{3}} x^{3} = 9 x^{3}$

Step 5.4.2.1.2

Rewrite the expression.

$32 = 9 x^{3}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $9 x^{3} = 32$ .

$9 x^{3} = 32$

Step 5.5.2

Divide each term in $9 x^{3} = 32$ by $9$ and simplify.

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Step 5.5.2.1

Divide each term in $9 x^{3} = 32$ by $9$ .

$\frac{9 x^{3}}{9} = \frac{32}{9}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $9$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{9 x^{3}}{9} = \frac{32}{9}$

Step 5.5.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{32}{9}$

Step 5.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{32}{9}}$

Step 5.5.4

Simplify $\sqrt[3]{\frac{32}{9}}$ .

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Step 5.5.4.1

Rewrite $\sqrt[3]{\frac{32}{9}}$ as $\frac{\sqrt[3]{32}}{\sqrt[3]{9}}$ .

$x = \frac{\sqrt[3]{32}}{\sqrt[3]{9}}$

Step 5.5.4.2

Simplify the numerator.

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Step 5.5.4.2.1

Rewrite $32$ as $2^{3} \cdot 4$ .

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Step 5.5.4.2.1.1

Factor $8$ out of $32$ .

$x = \frac{\sqrt[3]{8 (4)}}{\sqrt[3]{9}}$

Step 5.5.4.2.1.2

Rewrite $8$ as $2^{3}$ .

$x = \frac{\sqrt[3]{2^{3} \cdot 4}}{\sqrt[3]{9}}$

Step 5.5.4.2.2

Pull terms out from under the radical.

$x = \frac{2 \sqrt[3]{4}}{\sqrt[3]{9}}$

Step 5.5.4.3

Multiply $\frac{2 \sqrt[3]{4}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$x = \frac{2 \sqrt[3]{4}}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 5.5.4.4

Combine and simplify the denominator.

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Step 5.5.4.4.1

Multiply $\frac{2 \sqrt[3]{4}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 5.5.4.4.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 5.5.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 5.5.4.4.4

Add $1$ and $2$ .

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 5.5.4.4.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 5.5.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 5.5.4.4.5.2

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

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 5.5.4.4.5.3

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

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 5.5.4.4.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.4.5.4.1

Cancel the common factor.

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 5.5.4.4.5.4.2

Rewrite the expression.

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{9^{1}}$

Step 5.5.4.4.5.5

Evaluate the exponent.

$x = \frac{2 \sqrt[3]{4} {\sqrt[3]{9}}^{2}}{9}$

Step 5.5.4.5

Simplify the numerator.

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Step 5.5.4.5.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$x = \frac{2 \sqrt[3]{4} \sqrt[3]{9^{2}}}{9}$

Step 5.5.4.5.2

Raise $9$ to the power of $2$ .

$x = \frac{2 \sqrt[3]{4} \sqrt[3]{81}}{9}$

Step 5.5.4.5.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 5.5.4.5.3.1

Factor $27$ out of $81$ .

$x = \frac{2 \sqrt[3]{4} \sqrt[3]{27 (3)}}{9}$

Step 5.5.4.5.3.2

Rewrite $27$ as $3^{3}$ .

$x = \frac{2 \sqrt[3]{4} \sqrt[3]{3^{3} \cdot 3}}{9}$

Step 5.5.4.5.4

Pull terms out from under the radical.

$x = \frac{2 \sqrt[3]{4} \cdot 3 \sqrt[3]{3}}{9}$

Step 5.5.4.5.5

Combine exponents.

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Step 5.5.4.5.5.1

Multiply $3$ by $2$ .

$x = \frac{6 \sqrt[3]{4} \sqrt[3]{3}}{9}$

Step 5.5.4.5.5.2

Combine using the product rule for radicals.

$x = \frac{6 \sqrt[3]{3 \cdot 4}}{9}$

Step 5.5.4.5.5.3

Multiply $3$ by $4$ .

$x = \frac{6 \sqrt[3]{12}}{9}$

Step 5.5.4.6

Cancel the common factor of $6$ and $9$ .

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Step 5.5.4.6.1

Factor $3$ out of $6 \sqrt[3]{12}$ .

$x = \frac{3 (2 \sqrt[3]{12})}{9}$

Step 5.5.4.6.2

Cancel the common factors.

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Step 5.5.4.6.2.1

Factor $3$ out of $9$ .

$x = \frac{3 (2 \sqrt[3]{12})}{3 (3)}$

Step 5.5.4.6.2.2

Cancel the common factor.

$x = \frac{3 (2 \sqrt[3]{12})}{3 \cdot 3}$

Step 5.5.4.6.2.3

Rewrite the expression.

$x = \frac{2 \sqrt[3]{12}}{3}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{32}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 6.2.2

Simplify $\sqrt[3]{0}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{2 \sqrt[3]{12}}{3}$

Step 8

Evaluate the second derivative at $x = \frac{2 \sqrt[3]{12}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{96}{{(\frac{2 \sqrt[3]{12}}{3})}^{4}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Apply the product rule to $\frac{2 \sqrt[3]{12}}{3}$ .

$- \frac{96}{\frac{{(2 \sqrt[3]{12})}^{4}}{3^{4}}}$

Step 9.1.2

Simplify the numerator.

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Step 9.1.2.1

Apply the product rule to $2 \sqrt[3]{12}$ .

$- \frac{96}{\frac{2^{4} {\sqrt[3]{12}}^{4}}{3^{4}}}$

Step 9.1.2.2

Raise $2$ to the power of $4$ .

$- \frac{96}{\frac{16 {\sqrt[3]{12}}^{4}}{3^{4}}}$

Step 9.1.2.3

Rewrite ${\sqrt[3]{12}}^{4}$ as $\sqrt[3]{12^{4}}$ .

$- \frac{96}{\frac{16 \sqrt[3]{12^{4}}}{3^{4}}}$

Step 9.1.2.4

Raise $12$ to the power of $4$ .

$- \frac{96}{\frac{16 \sqrt[3]{20736}}{3^{4}}}$

Step 9.1.2.5

Rewrite $20736$ as $12^{3} \cdot 12$ .

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Step 9.1.2.5.1

Factor $1728$ out of $20736$ .

$- \frac{96}{\frac{16 \sqrt[3]{1728 (12)}}{3^{4}}}$

Step 9.1.2.5.2

Rewrite $1728$ as $12^{3}$ .

$- \frac{96}{\frac{16 \sqrt[3]{12^{3} \cdot 12}}{3^{4}}}$

Step 9.1.2.6

Pull terms out from under the radical.

$- \frac{96}{\frac{16 \cdot 12 \sqrt[3]{12}}{3^{4}}}$

Step 9.1.2.7

Multiply $16$ by $12$ .

$- \frac{96}{\frac{192 \sqrt[3]{12}}{3^{4}}}$

Step 9.1.3

Raise $3$ to the power of $4$ .

$- \frac{96}{\frac{192 \sqrt[3]{12}}{81}}$

Step 9.1.4

Cancel the common factor of $192$ and $81$ .

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Step 9.1.4.1

Factor $3$ out of $192 \sqrt[3]{12}$ .

$- \frac{96}{\frac{3 (64 \sqrt[3]{12})}{81}}$

Step 9.1.4.2

Cancel the common factors.

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Step 9.1.4.2.1

Factor $3$ out of $81$ .

$- \frac{96}{\frac{3 (64 \sqrt[3]{12})}{3 (27)}}$

Step 9.1.4.2.2

Cancel the common factor.

$- \frac{96}{\frac{3 (64 \sqrt[3]{12})}{3 \cdot 27}}$

Step 9.1.4.2.3

Rewrite the expression.

$- \frac{96}{\frac{64 \sqrt[3]{12}}{27}}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$- (96 \frac{27}{64 \sqrt[3]{12}})$

Step 9.3

Cancel the common factor of $32$ .

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Step 9.3.1

Factor $32$ out of $96$ .

$- (32 (3) \frac{27}{64 \sqrt[3]{12}})$

Step 9.3.2

Factor $32$ out of $64 \sqrt[3]{12}$ .

$- (32 (3) \frac{27}{32 (2 \sqrt[3]{12})})$

Step 9.3.3

Cancel the common factor.

$- (32 \cdot 3 \frac{27}{32 (2 \sqrt[3]{12})})$

Step 9.3.4

Rewrite the expression.

$- (3 \frac{27}{2 \sqrt[3]{12}})$

Step 9.4

Combine $3$ and $\frac{27}{2 \sqrt[3]{12}}$ .

$- \frac{3 \cdot 27}{2 \sqrt[3]{12}}$

Step 9.5

Multiply $3$ by $27$ .

$- \frac{81}{2 \sqrt[3]{12}}$

Step 9.6

Multiply $\frac{81}{2 \sqrt[3]{12}}$ by $\frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}}$ .

$- (\frac{81}{2 \sqrt[3]{12}} \cdot \frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}})$

Step 9.7

Simplify terms.

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Step 9.7.1

Combine and simplify the denominator.

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Step 9.7.1.1

Multiply $\frac{81}{2 \sqrt[3]{12}}$ by $\frac{{\sqrt[3]{12}}^{2}}{{\sqrt[3]{12}}^{2}}$ .

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \sqrt[3]{12} {\sqrt[3]{12}}^{2}}$

Step 9.7.1.2

Move $\sqrt[3]{12}$ .

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 (\sqrt[3]{12} {\sqrt[3]{12}}^{2})}$

Step 9.7.1.3

Raise $\sqrt[3]{12}$ to the power of $1$ .

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 ({\sqrt[3]{12}}^{1} {\sqrt[3]{12}}^{2})}$

Step 9.7.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 {\sqrt[3]{12}}^{1 + 2}}$

Step 9.7.1.5

Add $1$ and $2$ .

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 {\sqrt[3]{12}}^{3}}$

Step 9.7.1.6

Rewrite ${\sqrt[3]{12}}^{3}$ as $12$ .

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Step 9.7.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{12}$ as $12^{\frac{1}{3}}$ .

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 {(12^{\frac{1}{3}})}^{3}}$

Step 9.7.1.6.2

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

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \cdot 12^{\frac{1}{3} \cdot 3}}$

Step 9.7.1.6.3

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

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \cdot 12^{\frac{3}{3}}}$

Step 9.7.1.6.4

Cancel the common factor of $3$ .

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Step 9.7.1.6.4.1

Cancel the common factor.

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \cdot 12^{\frac{3}{3}}}$

Step 9.7.1.6.4.2

Rewrite the expression.

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \cdot 12^{1}}$

Step 9.7.1.6.5

Evaluate the exponent.

$- \frac{81 {\sqrt[3]{12}}^{2}}{2 \cdot 12}$

Step 9.7.2

Cancel the common factor of $81$ and $12$ .

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Step 9.7.2.1

Factor $3$ out of $81 {\sqrt[3]{12}}^{2}$ .

$- \frac{3 (27 {\sqrt[3]{12}}^{2})}{2 \cdot 12}$

Step 9.7.2.2

Cancel the common factors.

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Step 9.7.2.2.1

Factor $3$ out of $2 \cdot 12$ .

$- \frac{3 (27 {\sqrt[3]{12}}^{2})}{3 (2 \cdot 4)}$

Step 9.7.2.2.2

Cancel the common factor.

$- \frac{3 (27 {\sqrt[3]{12}}^{2})}{3 (2 \cdot 4)}$

Step 9.7.2.2.3

Rewrite the expression.

$- \frac{27 {\sqrt[3]{12}}^{2}}{2 \cdot 4}$

Step 9.8

Simplify the numerator.

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Step 9.8.1

Rewrite ${\sqrt[3]{12}}^{2}$ as $\sqrt[3]{12^{2}}$ .

$- \frac{27 \sqrt[3]{12^{2}}}{2 \cdot 4}$

Step 9.8.2

Raise $12$ to the power of $2$ .

$- \frac{27 \sqrt[3]{144}}{2 \cdot 4}$

Step 9.8.3

Rewrite $144$ as $2^{3} \cdot 18$ .

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Step 9.8.3.1

Factor $8$ out of $144$ .

$- \frac{27 \sqrt[3]{8 (18)}}{2 \cdot 4}$

Step 9.8.3.2

Rewrite $8$ as $2^{3}$ .

$- \frac{27 \sqrt[3]{2^{3} \cdot 18}}{2 \cdot 4}$

Step 9.8.4

Pull terms out from under the radical.

$- \frac{27 \cdot 2 \sqrt[3]{18}}{2 \cdot 4}$

Step 9.8.5

Multiply $27$ by $2$ .

$- \frac{54 \sqrt[3]{18}}{2 \cdot 4}$

Step 9.9

Reduce the expression by cancelling the common factors.

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Step 9.9.1

Multiply $2$ by $4$ .

$- \frac{54 \sqrt[3]{18}}{8}$

Step 9.9.2

Cancel the common factor of $54$ and $8$ .

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Step 9.9.2.1

Factor $2$ out of $54 \sqrt[3]{18}$ .

$- \frac{2 (27 \sqrt[3]{18})}{8}$

Step 9.9.2.2

Cancel the common factors.

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Step 9.9.2.2.1

Factor $2$ out of $8$ .

$- \frac{2 (27 \sqrt[3]{18})}{2 (4)}$

Step 9.9.2.2.2

Cancel the common factor.

$- \frac{2 (27 \sqrt[3]{18})}{2 \cdot 4}$

Step 9.9.2.2.3

Rewrite the expression.

$- \frac{27 \sqrt[3]{18}}{4}$

Step 10

$x = \frac{2 \sqrt[3]{12}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{2 \sqrt[3]{12}}{3}$ is a local maximum

Step 11

Find the y-value when $x = \frac{2 \sqrt[3]{12}}{3}$ .

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Step 11.1

Replace the variable $x$ with $\frac{2 \sqrt[3]{12}}{3}$ in the expression.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 9 \frac{2 \sqrt[3]{12}}{3} - \frac{16}{{(\frac{2 \sqrt[3]{12}}{3})}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Cancel the common factor of $3$ .

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Step 11.2.1.1.1

Factor $3$ out of $- 9$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 + 3 (- 3) (\frac{2 \sqrt[3]{12}}{3}) - \frac{16}{{(\frac{2 \sqrt[3]{12}}{3})}^{2}}$

Step 11.2.1.1.2

Cancel the common factor.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 + 3 \cdot (- 3 \frac{2 \sqrt[3]{12}}{3}) - \frac{16}{{(\frac{2 \sqrt[3]{12}}{3})}^{2}}$

Step 11.2.1.1.3

Rewrite the expression.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 3 (2 \sqrt[3]{12}) - \frac{16}{{(\frac{2 \sqrt[3]{12}}{3})}^{2}}$

Step 11.2.1.2

Multiply $2$ by $- 3$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{{(\frac{2 \sqrt[3]{12}}{3})}^{2}}$

Step 11.2.1.3

Simplify the denominator.

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Step 11.2.1.3.1

Apply the product rule to $\frac{2 \sqrt[3]{12}}{3}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{{(2 \sqrt[3]{12})}^{2}}{3^{2}}}$

Step 11.2.1.3.2

Simplify the numerator.

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Step 11.2.1.3.2.1

Apply the product rule to $2 \sqrt[3]{12}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{2^{2} {\sqrt[3]{12}}^{2}}{3^{2}}}$

Step 11.2.1.3.2.2

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

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 {\sqrt[3]{12}}^{2}}{3^{2}}}$

Step 11.2.1.3.2.3

Rewrite ${\sqrt[3]{12}}^{2}$ as $\sqrt[3]{12^{2}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 \sqrt[3]{12^{2}}}{3^{2}}}$

Step 11.2.1.3.2.4

Raise $12$ to the power of $2$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 \sqrt[3]{144}}{3^{2}}}$

Step 11.2.1.3.2.5

Rewrite $144$ as $2^{3} \cdot 18$ .

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Step 11.2.1.3.2.5.1

Factor $8$ out of $144$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 \sqrt[3]{8 (18)}}{3^{2}}}$

Step 11.2.1.3.2.5.2

Rewrite $8$ as $2^{3}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 \sqrt[3]{2^{3} \cdot 18}}{3^{2}}}$

Step 11.2.1.3.2.6

Pull terms out from under the radical.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{4 \cdot (2 \sqrt[3]{18})}{3^{2}}}$

Step 11.2.1.3.2.7

Multiply $4$ by $2$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{8 \sqrt[3]{18}}{3^{2}}}$

Step 11.2.1.3.3

Raise $3$ to the power of $2$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{16}{\frac{8 \sqrt[3]{18}}{9}}$

Step 11.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (16 (\frac{9}{8 \sqrt[3]{18}}))$

Step 11.2.1.5

Cancel the common factor of $8$ .

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Step 11.2.1.5.1

Factor $8$ out of $16$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (8 (2) (\frac{9}{8 \sqrt[3]{18}}))$

Step 11.2.1.5.2

Factor $8$ out of $8 \sqrt[3]{18}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (8 (2) (\frac{9}{8 (\sqrt[3]{18})}))$

Step 11.2.1.5.3

Cancel the common factor.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (8 \cdot (2 (\frac{9}{8 \sqrt[3]{18}})))$

Step 11.2.1.5.4

Rewrite the expression.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (2 (\frac{9}{\sqrt[3]{18}}))$

Step 11.2.1.6

Combine $2$ and $\frac{9}{\sqrt[3]{18}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{2 \cdot 9}{\sqrt[3]{18}}$

Step 11.2.1.7

Multiply $2$ by $9$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18}{\sqrt[3]{18}}$

Step 11.2.1.8

Multiply $\frac{18}{\sqrt[3]{18}}$ by $\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (\frac{18}{\sqrt[3]{18}} \cdot \frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}})$

Step 11.2.1.9

Combine and simplify the denominator.

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Step 11.2.1.9.1

Multiply $\frac{18}{\sqrt[3]{18}}$ by $\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{\sqrt[3]{18} {\sqrt[3]{18}}^{2}}$

Step 11.2.1.9.2

Raise $\sqrt[3]{18}$ to the power of $1$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{\sqrt[3]{18} {\sqrt[3]{18}}^{2}}$

Step 11.2.1.9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{1 + 2}}$

Step 11.2.1.9.4

Add $1$ and $2$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{3}}$

Step 11.2.1.9.5

Rewrite ${\sqrt[3]{18}}^{3}$ as $18$ .

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Step 11.2.1.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{18}$ as $18^{\frac{1}{3}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{{(18^{\frac{1}{3}})}^{3}}$

Step 11.2.1.9.5.2

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

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18^{\frac{1}{3} \cdot 3}}$

Step 11.2.1.9.5.3

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

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18^{\frac{3}{3}}}$

Step 11.2.1.9.5.4

Cancel the common factor of $3$ .

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Step 11.2.1.9.5.4.1

Cancel the common factor.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18^{\frac{3}{3}}}$

Step 11.2.1.9.5.4.2

Rewrite the expression.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18}$

Step 11.2.1.9.5.5

Evaluate the exponent.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18}$

Step 11.2.1.10

Cancel the common factor of $18$ .

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Step 11.2.1.10.1

Cancel the common factor.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \frac{18 {\sqrt[3]{18}}^{2}}{18}$

Step 11.2.1.10.2

Divide ${\sqrt[3]{18}}^{2}$ by $1$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - {\sqrt[3]{18}}^{2}$

Step 11.2.1.11

Rewrite ${\sqrt[3]{18}}^{2}$ as $\sqrt[3]{18^{2}}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \sqrt[3]{18^{2}}$

Step 11.2.1.12

Raise $18$ to the power of $2$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \sqrt[3]{324}$

Step 11.2.1.13

Rewrite $324$ as $3^{3} \cdot 12$ .

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Step 11.2.1.13.1

Factor $27$ out of $324$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \sqrt[3]{27 (12)}$

Step 11.2.1.13.2

Rewrite $27$ as $3^{3}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - \sqrt[3]{3^{3} \cdot 12}$

Step 11.2.1.14

Pull terms out from under the radical.

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - (3 \sqrt[3]{12})$

Step 11.2.1.15

Multiply $3$ by $- 1$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 6 \sqrt[3]{12} - 3 \sqrt[3]{12}$

Step 11.2.2

Subtract $3 \sqrt[3]{12}$ from $- 6 \sqrt[3]{12}$ .

$f (\frac{2 \sqrt[3]{12}}{3}) = 9 - 9 \sqrt[3]{12}$

Step 11.2.3

The final answer is $9 - 9 \sqrt[3]{12}$ .

$y = 9 - 9 \sqrt[3]{12}$

Step 12

These are the local extrema for $f (x) = 9 - 9 x - \frac{16}{x^{2}}$ .

$(\frac{2 \sqrt[3]{12}}{3}, 9 - 9 \sqrt[3]{12})$ is a local maxima

Step 13