Calculus Examples

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

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

$1 - - x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [\frac{2}{x^{3}}]$

Step 1.2.5

Multiply $- 1$ by $- 1$ .

$1 + 1 x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [\frac{2}{x^{3}}]$

Step 1.2.6

Multiply $x^{- 2}$ by $1$ .

$1 + x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [\frac{2}{x^{3}}]$

Step 1.2.7

Multiply $\frac{1}{x}$ by $0$ .

$1 + x^{- 2} + 0 + \frac{d}{d x} [\frac{2}{x^{3}}]$

Step 1.2.8

Add $x^{- 2}$ and $0$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

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 $g (x) = x^{3}$ .

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Step 1.3.3.1

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

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

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 $n = - 1$ .

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

Step 1.3.3.3

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

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

Step 1.3.4

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

$1 + x^{- 2} + 2 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

Multiply $3$ by $- 2$ .

$1 + x^{- 2} + 2 (- x^{- 6} (3 x^{2}))$

Step 1.3.6

Multiply $3$ by $- 1$ .

$1 + x^{- 2} + 2 (- 3 x^{- 6} x^{2})$

Step 1.3.7

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

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Step 1.3.7.1

Move $x^{2}$ .

$1 + x^{- 2} + 2 (- 3 (x^{2} x^{- 6}))$

Step 1.3.7.2

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

$1 + x^{- 2} + 2 (- 3 x^{2 - 6})$

Step 1.3.7.3

Subtract $6$ from $2$ .

$1 + x^{- 2} + 2 (- 3 x^{- 4})$

Step 1.3.8

Multiply $- 3$ by $2$ .

$1 + x^{- 2} - 6 x^{- 4}$

Step 1.4

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

$1 + \frac{1}{x^{2}} - 6 x^{- 4}$

Step 1.5

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

$1 + \frac{1}{x^{2}} - 6 \frac{1}{x^{4}}$

Step 1.6

Simplify.

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Step 1.6.1

Combine terms.

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Step 1.6.1.1

Combine $- 6$ and $\frac{1}{x^{4}}$ .

$1 + \frac{1}{x^{2}} + \frac{- 6}{x^{4}}$

Step 1.6.1.2

Move the negative in front of the fraction.

$1 + \frac{1}{x^{2}} - \frac{6}{x^{4}}$

Step 1.6.2

Reorder terms.

$\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{6}{x^{4}}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{d}{d x} (\frac{1}{x^{2}}) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{1}{x^{2}}]$ .

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Step 2.2.1

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

$f'' (x) = \frac{d}{d x} ({(x^{2})}^{- 1}) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.2

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

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Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$f'' (x) = \frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.2.2

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

$f'' (x) = - {u_{1}}^{- 2} \frac{d}{d x} x^{2} + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$f'' (x) = - {(x^{2})}^{- 2} \frac{d}{d x} x^{2} + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.3

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

$f'' (x) = - {(x^{2})}^{- 2} (2 x) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.4

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

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Step 2.2.4.1

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

$f'' (x) = - x^{2 \cdot - 2} (2 x) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.4.2

Multiply $2$ by $- 2$ .

$f'' (x) = - x^{- 4} (2 x) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.5

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 x^{- 4} x + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.6

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

$f'' (x) = - 2 (x \cdot x^{- 4}) + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.7

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

$f'' (x) = - 2 x^{1 - 4} + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.2.8

Subtract $4$ from $1$ .

$f'' (x) = - 2 x^{- 3} + \frac{d}{d x} (- \frac{6}{x^{4}}) + \frac{d}{d x} (1)$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{6}{x^{4}}]$ .

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Step 2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{x^{4}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$f'' (x) = - 2 x^{- 3} - 6 \frac{d}{d x} \frac{1}{x^{4}} + \frac{d}{d x} (1)$

Step 2.3.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$f'' (x) = - 2 x^{- 3} - 6 \frac{d}{d x} {(x^{4})}^{- 1} + \frac{d}{d x} (1)$

Step 2.3.3

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 $g (x) = x^{4}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

$f'' (x) = - 2 x^{- 3} - 6 (\frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} (x^{4})) + \frac{d}{d x} (1)$

Step 2.3.3.2

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

$f'' (x) = - 2 x^{- 3} - 6 (- {u_{2}}^{- 2} \frac{d}{d x} x^{4}) + \frac{d}{d x} (1)$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

$f'' (x) = - 2 x^{- 3} - 6 (- {(x^{4})}^{- 2} \frac{d}{d x} x^{4}) + \frac{d}{d x} (1)$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = - 2 x^{- 3} - 6 (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} (1)$

Step 2.3.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = - 2 x^{- 3} - 6 (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} (1)$

Step 2.3.5.2

Multiply $4$ by $- 2$ .

$f'' (x) = - 2 x^{- 3} - 6 (- x^{- 8} (4 x^{3})) + \frac{d}{d x} (1)$

Step 2.3.6

Multiply $4$ by $- 1$ .

$f'' (x) = - 2 x^{- 3} - 6 (- 4 x^{- 8} x^{3}) + \frac{d}{d x} (1)$

Step 2.3.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 2.3.7.1

Move $x^{3}$ .

$f'' (x) = - 2 x^{- 3} - 6 (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} (1)$

Step 2.3.7.2

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

$f'' (x) = - 2 x^{- 3} - 6 (- 4 x^{3 - 8}) + \frac{d}{d x} (1)$

Step 2.3.7.3

Subtract $8$ from $3$ .

$f'' (x) = - 2 x^{- 3} - 6 (- 4 x^{- 5}) + \frac{d}{d x} (1)$

Step 2.3.8

Multiply $- 4$ by $- 6$ .

$f'' (x) = - 2 x^{- 3} + 24 x^{- 5} + \frac{d}{d x} (1)$

Step 2.4

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

$f'' (x) = - 2 x^{- 3} + 24 x^{- 5} + 0$

Step 2.5

Simplify.

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Step 2.5.1

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

$f'' (x) = - 2 \frac{1}{x^{3}} + 24 x^{- 5} + 0$

Step 2.5.2

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

$f'' (x) = - 2 \frac{1}{x^{3}} + 24 (\frac{1}{x^{5}}) + 0$

Step 2.5.3

Combine terms.

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Step 2.5.3.1

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

$f'' (x) = \frac{- 2}{x^{3}} + 24 (\frac{1}{x^{5}}) + 0$

Step 2.5.3.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{2}{x^{3}} + 24 (\frac{1}{x^{5}}) + 0$

Step 2.5.3.3

Combine $24$ and $\frac{1}{x^{5}}$ .

$f'' (x) = - \frac{2}{x^{3}} + \frac{24}{x^{5}} + 0$

Step 2.5.3.4

Add $- \frac{2}{x^{3}} + \frac{24}{x^{5}}$ and $0$ .

$f'' (x) = - \frac{2}{x^{3}} + \frac{24}{x^{5}}$

Step 3

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

$\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1 = 0$

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

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

Step 4.1.1.2

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

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

Step 4.1.2

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

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Step 4.1.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

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

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

Step 4.1.2.5

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2.6

Multiply $x^{- 2}$ by $1$ .

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

Step 4.1.2.7

Multiply $\frac{1}{x}$ by $0$ .

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

Step 4.1.2.8

Add $x^{- 2}$ and $0$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

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 $g (x) = x^{3}$ .

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Step 4.1.3.3.1

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

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

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 $n = - 1$ .

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

Step 4.1.3.3.3

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

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

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 $n = 3$ .

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

Step 4.1.3.5

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

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Step 4.1.3.5.1

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

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

Step 4.1.3.5.2

Multiply $3$ by $- 2$ .

$f' (x) = 1 + x^{- 2} + 2 (- x^{- 6} (3 x^{2}))$

Step 4.1.3.6

Multiply $3$ by $- 1$ .

$f' (x) = 1 + x^{- 2} + 2 (- 3 x^{- 6} x^{2})$

Step 4.1.3.7

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

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Step 4.1.3.7.1

Move $x^{2}$ .

$f' (x) = 1 + x^{- 2} + 2 (- 3 (x^{2} x^{- 6}))$

Step 4.1.3.7.2

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

$f' (x) = 1 + x^{- 2} + 2 (- 3 x^{2 - 6})$

Step 4.1.3.7.3

Subtract $6$ from $2$ .

$f' (x) = 1 + x^{- 2} + 2 (- 3 x^{- 4})$

Step 4.1.3.8

Multiply $- 3$ by $2$ .

$f' (x) = 1 + x^{- 2} - 6 x^{- 4}$

Step 4.1.4

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

$f' (x) = 1 + \frac{1}{x^{2}} - 6 x^{- 4}$

Step 4.1.5

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

$f' (x) = 1 + \frac{1}{x^{2}} - 6 \frac{1}{x^{4}}$

Step 4.1.6

Simplify.

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Step 4.1.6.1

Combine terms.

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Step 4.1.6.1.1

Combine $- 6$ and $\frac{1}{x^{4}}$ .

$f' (x) = 1 + \frac{1}{x^{2}} + \frac{- 6}{x^{4}}$

Step 4.1.6.1.2

Move the negative in front of the fraction.

$f' (x) = 1 + \frac{1}{x^{2}} - \frac{6}{x^{4}}$

Step 4.1.6.2

Reorder terms.

$f' (x) = \frac{1}{x^{2}} - \frac{6}{x^{4}} + 1$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1$ .

$\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1 = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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

$x^{2}, x^{4}, 1, 1$

Step 5.2.2

Since $x^{2}, x^{4}, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{2}, x^{4}$ .

Step 5.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 5.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 5.2.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 5.2.6

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 5.2.7

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 5.2.8

The LCM of $x^{2}, x^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x$

Step 5.2.9

Simplify $x \cdot x \cdot x \cdot x$ .

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Step 5.2.9.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x$

Step 5.2.9.2

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

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Step 5.2.9.2.1

Multiply $x^{2}$ by $x$ .

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Step 5.2.9.2.1.1

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

$x^{2} \cdot x^{1} \cdot x$

Step 5.2.9.2.1.2

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

$x^{2 + 1} \cdot x$

Step 5.2.9.2.2

Add $2$ and $1$ .

$x^{3} \cdot x$

Step 5.2.9.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 5.2.9.3.1

Multiply $x^{3}$ by $x$ .

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Step 5.2.9.3.1.1

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

$x^{3} \cdot x^{1}$

Step 5.2.9.3.1.2

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

$x^{3 + 1}$

Step 5.2.9.3.2

Add $3$ and $1$ .

$x^{4}$

Step 5.3

Multiply each term in $\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1 = 0$ by $x^{4}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{1}{x^{2}} - \frac{6}{x^{4}} + 1 = 0$ by $x^{4}$ .

$\frac{1}{x^{2}} x^{4} - \frac{6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.1.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{1}{x^{2}} (x^{2} x^{2}) - \frac{6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.1.2

Cancel the common factor.

$\frac{1}{x^{2}} (x^{2} x^{2}) - \frac{6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.1.3

Rewrite the expression.

$x^{2} - \frac{6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.2

Cancel the common factor of $x^{4}$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{6}{x^{4}}$ into the numerator.

$x^{2} + \frac{- 6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.2.2

Cancel the common factor.

$x^{2} + \frac{- 6}{x^{4}} x^{4} + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.2.3

Rewrite the expression.

$x^{2} - 6 + 1 x^{4} = 0 x^{4}$

Step 5.3.2.1.3

Multiply $x^{4}$ by $1$ .

$x^{2} - 6 + x^{4} = 0 x^{4}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x^{4}$ .

$x^{2} - 6 + x^{4} = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} + u - 6 = 0$

$u = x^{2}$

Step 5.4.2

Factor $u^{2} + u - 6$ using the AC method.

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Step 5.4.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 5.4.2.2

Write the factored form using these integers.

$(u - 2) (u + 3) = 0$

Step 5.4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 2 = 0$

$u + 3 = 0$

Step 5.4.4

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 5.4.4.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 5.4.4.2

Add $2$ to both sides of the equation.

$u = 2$

Step 5.4.5

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 5.4.5.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 5.4.5.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 5.4.6

The final solution is all the values that make $(u - 2) (u + 3) = 0$ true.

$u = 2, - 3$

Step 5.4.7

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 2$

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

Step 5.4.8

Solve the first equation for $x$ .

$x^{2} = 2$

Step 5.4.9

Solve the equation for $x$ .

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Step 5.4.9.1

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

$x = \pm \sqrt{2}$

Step 5.4.9.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.4.9.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 5.4.9.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 5.4.9.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 5.4.10

Solve the second equation for $x$ .

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

Step 5.4.11

Solve the equation for $x$ .

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Step 5.4.11.1

Remove parentheses.

$x^{2} = - 3$

Step 5.4.11.2

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

$x = \pm \sqrt{- 3}$

Step 5.4.11.3

Simplify $\pm \sqrt{- 3}$ .

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Step 5.4.11.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 5.4.11.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 5.4.11.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{3}$

Step 5.4.11.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.4.11.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{3}$

Step 5.4.11.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{3}$

Step 5.4.11.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{3}, - i \sqrt{3}$

Step 5.4.12

The solution to $x^{4} + x^{2} - 6 = 0$ is $x = \sqrt{2}, - \sqrt{2}, i \sqrt{3}, - i \sqrt{3}$ .

$x = \sqrt{2}, - \sqrt{2}, i \sqrt{3}, - i \sqrt{3}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

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.

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3

Set the denominator in $\frac{6}{x^{4}}$ equal to $0$ to find where the expression is undefined.

$x^{4} = 0$

Step 6.4

Solve for $x$ .

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Step 6.4.1

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

$x = \pm \sqrt[4]{0}$

Step 6.4.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 6.4.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 6.4.2.2

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

$x = \pm 0$

Step 6.4.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = \sqrt{2}, - \sqrt{2}$

Step 8

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

$- \frac{2}{{(\sqrt{2})}^{3}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$- \frac{2}{\sqrt{2^{3}}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.1.2

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

$- \frac{2}{\sqrt{8}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.1.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.1.1.3.1

Factor $4$ out of $8$ .

$- \frac{2}{\sqrt{4 (2)}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.1.3.2

Rewrite $4$ as $2^{2}$ .

$- \frac{2}{\sqrt{2^{2} \cdot 2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.1.4

Pull terms out from under the radical.

$- \frac{2}{2 \sqrt{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.2

Cancel the common factor of $2$ .

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Step 9.1.2.1

Cancel the common factor.

$- \frac{2}{2 \sqrt{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.2.2

Rewrite the expression.

$- \frac{1}{\sqrt{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4

Combine and simplify the denominator.

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Step 9.1.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.4

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

$- \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.5

Add $1$ and $1$ .

$- \frac{\sqrt{2}}{{\sqrt{2}}^{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.1.4.6.1

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

$- \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6.2

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

$- \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6.4

Cancel the common factor of $2$ .

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Step 9.1.4.6.4.1

Cancel the common factor.

$- \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6.4.2

Rewrite the expression.

$- \frac{\sqrt{2}}{2^{1}} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.4.6.5

Evaluate the exponent.

$- \frac{\sqrt{2}}{2} + \frac{24}{{(\sqrt{2})}^{5}}$

Step 9.1.5

Simplify the denominator.

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Step 9.1.5.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$- \frac{\sqrt{2}}{2} + \frac{24}{\sqrt{2^{5}}}$

Step 9.1.5.2

Raise $2$ to the power of $5$ .

$- \frac{\sqrt{2}}{2} + \frac{24}{\sqrt{32}}$

Step 9.1.5.3

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

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Step 9.1.5.3.1

Factor $16$ out of $32$ .

$- \frac{\sqrt{2}}{2} + \frac{24}{\sqrt{16 (2)}}$

Step 9.1.5.3.2

Rewrite $16$ as $4^{2}$ .

$- \frac{\sqrt{2}}{2} + \frac{24}{\sqrt{4^{2} \cdot 2}}$

Step 9.1.5.4

Pull terms out from under the radical.

$- \frac{\sqrt{2}}{2} + \frac{24}{4 \sqrt{2}}$

Step 9.1.6

Cancel the common factor of $24$ and $4$ .

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Step 9.1.6.1

Factor $4$ out of $24$ .

$- \frac{\sqrt{2}}{2} + \frac{4 \cdot 6}{4 \sqrt{2}}$

Step 9.1.6.2

Cancel the common factors.

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Step 9.1.6.2.1

Factor $4$ out of $4 \sqrt{2}$ .

$- \frac{\sqrt{2}}{2} + \frac{4 \cdot 6}{4 (\sqrt{2})}$

Step 9.1.6.2.2

Cancel the common factor.

$- \frac{\sqrt{2}}{2} + \frac{4 \cdot 6}{4 \sqrt{2}}$

Step 9.1.6.2.3

Rewrite the expression.

$- \frac{\sqrt{2}}{2} + \frac{6}{\sqrt{2}}$

Step 9.1.7

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{\sqrt{2}}{2} + \frac{6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9.1.8

Combine and simplify the denominator.

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Step 9.1.8.1

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.1.8.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 9.1.8.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 9.1.8.4

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

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.1.8.5

Add $1$ and $1$ .

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 9.1.8.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.1.8.6.1

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

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.1.8.6.2

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

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.1.8.6.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.1.8.6.4

Cancel the common factor of $2$ .

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Step 9.1.8.6.4.1

Cancel the common factor.

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.1.8.6.4.2

Rewrite the expression.

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{2^{1}}$

Step 9.1.8.6.5

Evaluate the exponent.

$- \frac{\sqrt{2}}{2} + \frac{6 \sqrt{2}}{2}$

Step 9.1.9

Cancel the common factor of $6$ and $2$ .

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Step 9.1.9.1

Factor $2$ out of $6 \sqrt{2}$ .

$- \frac{\sqrt{2}}{2} + \frac{2 (3 \sqrt{2})}{2}$

Step 9.1.9.2

Cancel the common factors.

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Step 9.1.9.2.1

Factor $2$ out of $2$ .

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

Step 9.1.9.2.2

Cancel the common factor.

$- \frac{\sqrt{2}}{2} + \frac{2 (3 \sqrt{2})}{2 \cdot 1}$

Step 9.1.9.2.3

Rewrite the expression.

$- \frac{\sqrt{2}}{2} + \frac{3 \sqrt{2}}{1}$

Step 9.1.9.2.4

Divide $3 \sqrt{2}$ by $1$ .

$- \frac{\sqrt{2}}{2} + 3 \sqrt{2}$

Step 9.2

To write $3 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{\sqrt{2}}{2} + 3 \sqrt{2} \cdot \frac{2}{2}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $3 \sqrt{2}$ and $\frac{2}{2}$ .

$- \frac{\sqrt{2}}{2} + \frac{3 \sqrt{2} \cdot 2}{2}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{- \sqrt{2} + 3 \sqrt{2} \cdot 2}{2}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $2$ by $3$ .

$\frac{- \sqrt{2} + 6 \sqrt{2}}{2}$

Step 9.4.2

Add $- \sqrt{2}$ and $6 \sqrt{2}$ .

$\frac{5 \sqrt{2}}{2}$

Step 10

$x = \sqrt{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \sqrt{2}$ is a local minimum

Step 11

Find the y-value when $x = \sqrt{2}$ .

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Step 11.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

$f (\sqrt{2}) = (\sqrt{2}) - \frac{1}{\sqrt{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

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

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 11.2.1.2

Combine and simplify the denominator.

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Step 11.2.1.2.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.4

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.5

Add $1$ and $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 11.2.1.2.6.1

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.6.2

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

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

Step 11.2.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.2.6.4.1

Cancel the common factor.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.6.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.2.6.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{{(\sqrt{2})}^{3}}$

Step 11.2.1.3

Simplify the denominator.

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Step 11.2.1.3.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{\sqrt{2^{3}}}$

Step 11.2.1.3.2

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{\sqrt{8}}$

Step 11.2.1.3.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 11.2.1.3.3.1

Factor $4$ out of $8$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{\sqrt{4 (2)}}$

Step 11.2.1.3.3.2

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{\sqrt{2^{2} \cdot 2}}$

Step 11.2.1.3.4

Pull terms out from under the radical.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{2 \sqrt{2}}$

Step 11.2.1.4

Cancel the common factor of $2$ .

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Step 11.2.1.4.1

Cancel the common factor.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{2}{2 \sqrt{2}}$

Step 11.2.1.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{2}}$

Step 11.2.1.5

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 11.2.1.6

Combine and simplify the denominator.

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Step 11.2.1.6.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.2.1.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.2.1.6.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.2.1.6.4

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 11.2.1.6.5

Add $1$ and $1$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 11.2.1.6.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 11.2.1.6.6.1

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 11.2.1.6.6.2

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

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 11.2.1.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.2.1.6.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.6.6.4.1

Cancel the common factor.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.2.1.6.6.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 11.2.1.6.6.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 11.2.2

Simplify terms.

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Step 11.2.2.1

Combine the numerators over the common denominator.

$f (\sqrt{2}) = \sqrt{2} + \frac{- \sqrt{2} + \sqrt{2}}{2}$

Step 11.2.2.2

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$f (\sqrt{2}) = \sqrt{2} + \frac{0}{2}$

Step 11.2.2.3

Simplify the expression.

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Step 11.2.2.3.1

Divide $0$ by $2$ .

$f (\sqrt{2}) = \sqrt{2} + 0$

Step 11.2.2.3.2

Add $\sqrt{2}$ and $0$ .

$f (\sqrt{2}) = \sqrt{2}$

Step 11.2.3

The final answer is $\sqrt{2}$ .

$y = \sqrt{2}$

Step 12

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

$- \frac{2}{{(- \sqrt{2})}^{3}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Simplify the denominator.

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Step 13.1.1.1

Apply the product rule to $- \sqrt{2}$ .

$- \frac{2}{{(- 1)}^{3} {\sqrt{2}}^{3}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.2

Raise $- 1$ to the power of $3$ .

$- \frac{2}{- {\sqrt{2}}^{3}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.3

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$- \frac{2}{- \sqrt{2^{3}}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.4

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

$- \frac{2}{- \sqrt{8}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 13.1.1.5.1

Factor $4$ out of $8$ .

$- \frac{2}{- \sqrt{4 (2)}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.5.2

Rewrite $4$ as $2^{2}$ .

$- \frac{2}{- \sqrt{2^{2} \cdot 2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.6

Pull terms out from under the radical.

$- \frac{2}{- 1 \cdot 2 \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.1.7

Multiply $- 1$ by $2$ .

$- \frac{2}{- 2 \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.2

Cancel the common factor of $2$ and $- 2$ .

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Step 13.1.2.1

Factor $2$ out of $2$ .

$- \frac{2 (1)}{- 2 \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.2.2

Cancel the common factors.

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Step 13.1.2.2.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$- \frac{2 (1)}{2 (- \sqrt{2})} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.2.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 (- \sqrt{2})} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.2.2.3

Rewrite the expression.

$- \frac{1}{- \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 13.1.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$- \frac{- 1 (- 1)}{- \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.3.2

Move the negative in front of the fraction.

$- - \frac{1}{\sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.4

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5

Combine and simplify the denominator.

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Step 13.1.5.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$- - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$- - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.4

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

$- - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.5

Add $1$ and $1$ .

$- - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.1.5.6.1

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

$- - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6.2

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

$- - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$- - \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6.4

Cancel the common factor of $2$ .

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Step 13.1.5.6.4.1

Cancel the common factor.

$- - \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6.4.2

Rewrite the expression.

$- - \frac{\sqrt{2}}{2^{1}} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.5.6.5

Evaluate the exponent.

$- - \frac{\sqrt{2}}{2} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.6

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 13.1.6.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{2}}{2} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.6.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$\frac{\sqrt{2}}{2} + \frac{24}{{(- \sqrt{2})}^{5}}$

Step 13.1.7

Simplify the denominator.

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Step 13.1.7.1

Apply the product rule to $- \sqrt{2}$ .

$\frac{\sqrt{2}}{2} + \frac{24}{{(- 1)}^{5} {\sqrt{2}}^{5}}$

Step 13.1.7.2

Raise $- 1$ to the power of $5$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- {\sqrt{2}}^{5}}$

Step 13.1.7.3

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- \sqrt{2^{5}}}$

Step 13.1.7.4

Raise $2$ to the power of $5$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- \sqrt{32}}$

Step 13.1.7.5

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

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Step 13.1.7.5.1

Factor $16$ out of $32$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- \sqrt{16 (2)}}$

Step 13.1.7.5.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- \sqrt{4^{2} \cdot 2}}$

Step 13.1.7.6

Pull terms out from under the radical.

$\frac{\sqrt{2}}{2} + \frac{24}{- 1 \cdot 4 \sqrt{2}}$

Step 13.1.7.7

Multiply $- 1$ by $4$ .

$\frac{\sqrt{2}}{2} + \frac{24}{- 4 \sqrt{2}}$

Step 13.1.8

Cancel the common factor of $24$ and $- 4$ .

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Step 13.1.8.1

Factor $4$ out of $24$ .

$\frac{\sqrt{2}}{2} + \frac{4 (6)}{- 4 \sqrt{2}}$

Step 13.1.8.2

Cancel the common factors.

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Step 13.1.8.2.1

Factor $4$ out of $- 4 \sqrt{2}$ .

$\frac{\sqrt{2}}{2} + \frac{4 (6)}{4 (- \sqrt{2})}$

Step 13.1.8.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{2} + \frac{4 \cdot 6}{4 (- \sqrt{2})}$

Step 13.1.8.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{2} + \frac{6}{- \sqrt{2}}$

Step 13.1.9

Move the negative in front of the fraction.

$\frac{\sqrt{2}}{2} - \frac{6}{\sqrt{2}}$

Step 13.1.10

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{2} - (\frac{6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 13.1.11

Combine and simplify the denominator.

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Step 13.1.11.1

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 13.1.11.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 13.1.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 13.1.11.4

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

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 13.1.11.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 13.1.11.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.1.11.6.1

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

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 13.1.11.6.2

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

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 13.1.11.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 13.1.11.6.4

Cancel the common factor of $2$ .

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Step 13.1.11.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 13.1.11.6.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{2^{1}}$

Step 13.1.11.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2} - \frac{6 \sqrt{2}}{2}$

Step 13.1.12

Cancel the common factor of $6$ and $2$ .

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Step 13.1.12.1

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{\sqrt{2}}{2} - \frac{2 (3 \sqrt{2})}{2}$

Step 13.1.12.2

Cancel the common factors.

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Step 13.1.12.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt{2}}{2} - \frac{2 (3 \sqrt{2})}{2 (1)}$

Step 13.1.12.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{2} - \frac{2 (3 \sqrt{2})}{2 \cdot 1}$

Step 13.1.12.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{2} - \frac{3 \sqrt{2}}{1}$

Step 13.1.12.2.4

Divide $3 \sqrt{2}$ by $1$ .

$\frac{\sqrt{2}}{2} - (3 \sqrt{2})$

Step 13.1.13

Multiply $3$ by $- 1$ .

$\frac{\sqrt{2}}{2} - 3 \sqrt{2}$

Step 13.2

To write $- 3 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{2}}{2} - 3 \sqrt{2} \cdot \frac{2}{2}$

Step 13.3

Combine fractions.

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Step 13.3.1

Combine $- 3 \sqrt{2}$ and $\frac{2}{2}$ .

$\frac{\sqrt{2}}{2} + \frac{- 3 \sqrt{2} \cdot 2}{2}$

Step 13.3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} - 3 \sqrt{2} \cdot 2}{2}$

Step 13.4

Simplify the numerator.

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Step 13.4.1

Multiply $2$ by $- 3$ .

$\frac{\sqrt{2} - 6 \sqrt{2}}{2}$

Step 13.4.2

Subtract $6 \sqrt{2}$ from $\sqrt{2}$ .

$\frac{- 5 \sqrt{2}}{2}$

Step 13.5

Move the negative in front of the fraction.

$- \frac{5 \sqrt{2}}{2}$

Step 14

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

$x = - \sqrt{2}$ is a local maximum

Step 15

Find the y-value when $x = - \sqrt{2}$ .

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Step 15.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

$f (- \sqrt{2}) = (- \sqrt{2}) - \frac{1}{- \sqrt{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $1$ and $- 1$ .

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Step 15.2.1.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 15.2.1.1.2

Move the negative in front of the fraction.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{1}{\sqrt{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.2

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 15.2.1.3

Combine and simplify the denominator.

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Step 15.2.1.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.4

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.5

Add $1$ and $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{{\sqrt{2}}^{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.2.1.3.6.1

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.6.2

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

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

Step 15.2.1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 15.2.1.3.6.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2^{\frac{2}{2}}} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.6.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.3.6.5

Evaluate the exponent.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.4

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 15.2.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- \sqrt{2}) = - \sqrt{2} + 1 (\frac{\sqrt{2}}{2}) + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.4.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{{(- \sqrt{2})}^{3}}$

Step 15.2.1.5

Simplify the denominator.

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Step 15.2.1.5.1

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{{(- 1)}^{3} {\sqrt{2}}^{3}}$

Step 15.2.1.5.2

Raise $- 1$ to the power of $3$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- {\sqrt{2}}^{3}}$

Step 15.2.1.5.3

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- \sqrt{2^{3}}}$

Step 15.2.1.5.4

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- \sqrt{8}}$

Step 15.2.1.5.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 15.2.1.5.5.1

Factor $4$ out of $8$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- \sqrt{4 (2)}}$

Step 15.2.1.5.5.2

Rewrite $4$ as $2^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- \sqrt{2^{2} \cdot 2}}$

Step 15.2.1.5.6

Pull terms out from under the radical.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- 1 \cdot (2 \sqrt{2})}$

Step 15.2.1.5.7

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2}{- 2 \sqrt{2}}$

Step 15.2.1.6

Cancel the common factor of $2$ and $- 2$ .

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Step 15.2.1.6.1

Factor $2$ out of $2$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2 (1)}{- 2 \sqrt{2}}$

Step 15.2.1.6.2

Cancel the common factors.

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Step 15.2.1.6.2.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2 (1)}{2 (- \sqrt{2})}$

Step 15.2.1.6.2.2

Cancel the common factor.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{2 \cdot 1}{2 (- \sqrt{2})}$

Step 15.2.1.6.2.3

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{1}{- \sqrt{2}}$

Step 15.2.1.7

Cancel the common factor of $1$ and $- 1$ .

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Step 15.2.1.7.1

Rewrite $1$ as $- 1 (- 1)$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} + \frac{- 1 \cdot - 1}{- \sqrt{2}}$

Step 15.2.1.7.2

Move the negative in front of the fraction.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{1}{\sqrt{2}}$

Step 15.2.1.8

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 15.2.1.9

Combine and simplify the denominator.

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Step 15.2.1.9.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 15.2.1.9.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 15.2.1.9.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 15.2.1.9.4

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 15.2.1.9.5

Add $1$ and $1$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 15.2.1.9.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.2.1.9.6.1

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 15.2.1.9.6.2

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

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 15.2.1.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 15.2.1.9.6.4

Cancel the common factor of $2$ .

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Step 15.2.1.9.6.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 15.2.1.9.6.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$

Step 15.2.1.9.6.5

Evaluate the exponent.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$

Step 15.2.2

Simplify terms.

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Step 15.2.2.1

Combine the numerators over the common denominator.

$f (- \sqrt{2}) = - \sqrt{2} + \frac{\sqrt{2} - \sqrt{2}}{2}$

Step 15.2.2.2

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} + \frac{0}{2}$

Step 15.2.2.3

Simplify the expression.

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Step 15.2.2.3.1

Divide $0$ by $2$ .

$f (- \sqrt{2}) = - \sqrt{2} + 0$

Step 15.2.2.3.2

Add $- \sqrt{2}$ and $0$ .

$f (- \sqrt{2}) = - \sqrt{2}$

Step 15.2.3

The final answer is $- \sqrt{2}$ .

$y = - \sqrt{2}$

Step 16

These are the local extrema for $f (x) = x - \frac{1}{x} + \frac{2}{x^{3}}$ .

$(\sqrt{2}, \sqrt{2})$ is a local minima

$(- \sqrt{2}, - \sqrt{2})$ is a local maxima

Step 17