Calculus Examples

$y = \frac{x}{1 + x^{4}}$

Step 1

Write $y = \frac{x}{1 + x^{4}}$ as a function.

$f (x) = \frac{x}{1 + x^{4}}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 1 + x^{4}$ .

$\frac{(1 + x^{4}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

$\frac{(1 + x^{4}) \cdot 1 - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 2.2.2

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

$\frac{1 + x^{4} - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 2.2.3

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

$\frac{1 + x^{4} - x (\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 2.2.4

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

$\frac{1 + x^{4} - x (0 + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 2.2.5

Add $0$ and $\frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [x^{4}]}{{(1 + x^{4})}^{2}}$

Step 2.2.6

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

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

Step 2.2.7

Multiply $4$ by $- 1$ .

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

Step 2.3

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

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Step 2.3.1

Move $x^{3}$ .

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.3

Add $3$ and $1$ .

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

Step 2.4

Subtract $4 x^{4}$ from $x^{4}$ .

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

Step 2.5

Reorder terms.

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

Step 3.2

Differentiate.

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Step 3.2.1

Multiply the exponents in ${({(x^{4} + 1)}^{2})}^{2}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Multiply $2$ by $2$ .

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

Step 3.2.2

By the Sum Rule, the derivative of $- 3 x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [1]$ .

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

Step 3.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $4$ by $- 3$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3} + \frac{d}{d x} (1)) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 3.2.6

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

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3} + 0) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 3.2.7

Add $- 12 x^{3}$ and $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{4} + 1$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 3.3.2

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

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (2 u \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 3.3.3

Replace all occurrences of $u$ with $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (2 (x^{4} + 1) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 3.4

Differentiate.

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Step 3.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 3.4.2

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

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (1)))}{{(x^{4} + 1)}^{4}}$

Step 3.4.3

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

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3} + \frac{d}{d x} (1)))}{{(x^{4} + 1)}^{4}}$

Step 3.4.4

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

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3} + 0))}{{(x^{4} + 1)}^{4}}$

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

Add $4 x^{3}$ and $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3}))}{{(x^{4} + 1)}^{4}}$

Step 3.4.5.2

Move $4$ to the left of $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) (4 \cdot ((x^{4} + 1) x^{3}))}{{(x^{4} + 1)}^{4}}$

Step 3.4.5.3

Multiply $4$ by $- 2$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 8 (- 3 x^{4} + 1) ((x^{4} + 1) x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) + (- 8 (- 3 x^{4}) - 8 \cdot 1) ((x^{4} + 1) x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.2

Apply the distributive property.

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3

Simplify the numerator.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 12 {(x^{4} + 1)}^{2} x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.2

Rewrite ${(x^{4} + 1)}^{2}$ as $(x^{4} + 1) (x^{4} + 1)$ .

$f'' (x) = \frac{- 12 ((x^{4} + 1) (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.3

Expand $(x^{4} + 1) (x^{4} + 1)$ using the FOIL Method.

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Step 3.5.3.1.3.1

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} (x^{4} + 1) + 1 (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.3.2

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} x^{4} + x^{4} \cdot 1 + 1 (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.3.3

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} x^{4} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4

Simplify and combine like terms.

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Step 3.5.3.1.4.1

Simplify each term.

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Step 3.5.3.1.4.1.1

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 3.5.3.1.4.1.1.1

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

$f'' (x) = \frac{- 12 (x^{4 + 4} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4.1.1.2

Add $4$ and $4$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4.1.2

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

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4.1.3

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

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4.1.4

Multiply $1$ by $1$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + x^{4} + 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.4.2

Add $x^{4}$ and $x^{4}$ .

$f'' (x) = \frac{- 12 (x^{8} + 2 x^{4} + 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.5

Apply the distributive property.

$f'' (x) = \frac{(- 12 x^{8} - 12 (2 x^{4}) - 12 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.6

Simplify.

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Step 3.5.3.1.6.1

Multiply $2$ by $- 12$ .

$f'' (x) = \frac{(- 12 x^{8} - 24 x^{4} - 12 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.6.2

Multiply $- 12$ by $1$ .

$f'' (x) = \frac{(- 12 x^{8} - 24 x^{4} - 12) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.7

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{8} x^{3} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8

Simplify.

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Step 3.5.3.1.8.1

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

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Step 3.5.3.1.8.1.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 (x^{3} x^{8}) - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8.1.2

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

$f'' (x) = \frac{- 12 x^{3 + 8} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8.1.3

Add $3$ and $8$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.5.3.1.8.2.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 (x^{3} x^{4}) - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8.2.2

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

$f'' (x) = \frac{- 12 x^{11} - 24 x^{3 + 4} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.8.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.9

Simplify each term.

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Step 3.5.3.1.9.1

Multiply $- 3$ by $- 8$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.9.2

Multiply $- 8$ by $1$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.10

Simplify each term.

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Step 3.5.3.1.10.1

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.5.3.1.10.1.1

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

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{4 + 3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.10.1.2

Add $4$ and $3$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{7} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.10.2

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

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.11

Expand $(24 x^{4} - 8) (x^{7} + x^{3})$ using the FOIL Method.

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Step 3.5.3.1.11.1

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} (x^{7} + x^{3}) - 8 (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.11.2

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} x^{7} + 24 x^{4} x^{3} - 8 (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.11.3

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} x^{7} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12

Simplify and combine like terms.

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Step 3.5.3.1.12.1

Simplify each term.

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Step 3.5.3.1.12.1.1

Multiply $x^{4}$ by $x^{7}$ by adding the exponents.

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Step 3.5.3.1.12.1.1.1

Move $x^{7}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 (x^{7} x^{4}) + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.1.1.2

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

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{7 + 4} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.1.1.3

Add $7$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.5.3.1.12.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 (x^{3} x^{4}) - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.1.2.2

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

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{3 + 4} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.1.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{7} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.1.12.2

Subtract $8 x^{7}$ from $24 x^{7}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 16 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.2

Add $- 12 x^{11}$ and $24 x^{11}$ .

$f'' (x) = \frac{12 x^{11} - 24 x^{7} - 12 x^{3} + 16 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.3

Add $- 24 x^{7}$ and $16 x^{7}$ .

$f'' (x) = \frac{12 x^{11} - 8 x^{7} - 12 x^{3} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.3.4

Subtract $8 x^{3}$ from $- 12 x^{3}$ .

$f'' (x) = \frac{12 x^{11} - 8 x^{7} - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.4

Simplify the numerator.

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Step 3.5.4.1

Factor $4 x^{3}$ out of $12 x^{11} - 8 x^{7} - 20 x^{3}$ .

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Step 3.5.4.1.1

Factor $4 x^{3}$ out of $12 x^{11}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) - 8 x^{7} - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.4.1.2

Factor $4 x^{3}$ out of $- 8 x^{7}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4}) - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 3.5.4.1.3

Factor $4 x^{3}$ out of $- 20 x^{3}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4}) + 4 x^{3} (- 5)}{{(x^{4} + 1)}^{4}}$

Step 3.5.4.1.4

Factor $4 x^{3}$ out of $4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4})$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8} - 2 x^{4}) + 4 x^{3} (- 5)}{{(x^{4} + 1)}^{4}}$

Step 3.5.4.1.5

Factor $4 x^{3}$ out of $4 x^{3} (3 x^{8} - 2 x^{4}) + 4 x^{3} (- 5)$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8} - 2 x^{4} - 5)}{{(x^{4} + 1)}^{4}}$

Step 3.5.4.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

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

Step 3.5.4.3

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

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

Step 3.5.4.4

Factor by grouping.

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Step 3.5.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = - 2$ .

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Step 3.5.4.4.1.1

Factor $- 2$ out of $- 2 u$ .

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

Step 3.5.4.4.1.2

Rewrite $- 2$ as $3$ plus $- 5$

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

Step 3.5.4.4.1.3

Apply the distributive property.

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

Step 3.5.4.4.2

Factor out the greatest common factor from each group.

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Step 3.5.4.4.2.1

Group the first two terms and the last two terms.

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

Step 3.5.4.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.5.4.4.3

Factor the polynomial by factoring out the greatest common factor, $u + 1$ .

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

Step 3.5.4.5

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

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

Step 3.5.5

Cancel the common factor of $x^{4} + 1$ and ${(x^{4} + 1)}^{4}$ .

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Step 3.5.5.1

Factor $x^{4} + 1$ out of $4 x^{3} (x^{4} + 1) (3 x^{4} - 5)$ .

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

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $x^{4} + 1$ out of ${(x^{4} + 1)}^{4}$ .

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

Step 3.5.5.2.2

Cancel the common factor.

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

Step 3.5.5.2.3

Rewrite the expression.

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

Step 4

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

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$\frac{(1 + x^{4}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

$\frac{(1 + x^{4}) \cdot 1 - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 5.1.2.2

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

$\frac{1 + x^{4} - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 5.1.2.3

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

$\frac{1 + x^{4} - x (\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 5.1.2.4

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

$\frac{1 + x^{4} - x (0 + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 5.1.2.5

Add $0$ and $\frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [x^{4}]}{{(1 + x^{4})}^{2}}$

Step 5.1.2.6

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

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

Step 5.1.2.7

Multiply $4$ by $- 1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

Move $x^{3}$ .

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

Step 5.1.3.2

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

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Step 5.1.3.2.1

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

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

Step 5.1.3.2.2

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

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

Step 5.1.3.3

Add $3$ and $1$ .

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

Step 5.1.4

Subtract $4 x^{4}$ from $x^{4}$ .

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

Step 5.1.5

Reorder terms.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$ .

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

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$- 3 x^{4} + 1 = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Subtract $1$ from both sides of the equation.

$- 3 x^{4} = - 1$

Step 6.3.2

Divide each term in $- 3 x^{4} = - 1$ by $- 3$ and simplify.

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Step 6.3.2.1

Divide each term in $- 3 x^{4} = - 1$ by $- 3$ .

$\frac{- 3 x^{4}}{- 3} = \frac{- 1}{- 3}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{4}}{- 3} = \frac{- 1}{- 3}$

Step 6.3.2.2.1.2

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

$x^{4} = \frac{- 1}{- 3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Dividing two negative values results in a positive value.

$x^{4} = \frac{1}{3}$

Step 6.3.3

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

$x = \pm \sqrt[4]{\frac{1}{3}}$

Step 6.3.4

Simplify $\pm \sqrt[4]{\frac{1}{3}}$ .

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Step 6.3.4.1

Rewrite $\sqrt[4]{\frac{1}{3}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{3}}$ .

$x = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{3}}$

Step 6.3.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[4]{3}}$

Step 6.3.4.3

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

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

Step 6.3.4.4

Combine and simplify the denominator.

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Step 6.3.4.4.1

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

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

Step 6.3.4.4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

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

Step 6.3.4.4.3

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

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

Step 6.3.4.4.4

Add $1$ and $3$ .

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

Step 6.3.4.4.5

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

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Step 6.3.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

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

Step 6.3.4.4.5.2

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

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

Step 6.3.4.4.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 6.3.4.4.5.4

Cancel the common factor of $4$ .

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Step 6.3.4.4.5.4.1

Cancel the common factor.

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

Step 6.3.4.4.5.4.2

Rewrite the expression.

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

Step 6.3.4.4.5.5

Evaluate the exponent.

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

Step 6.3.4.5

Simplify the numerator.

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Step 6.3.4.5.1

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

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

Step 6.3.4.5.2

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

$x = \pm \frac{\sqrt[4]{27}}{3}$

Step 6.3.5

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

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Step 6.3.5.1

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

$x = \frac{\sqrt[4]{27}}{3}$

Step 6.3.5.2

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

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

Step 6.3.5.3

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

$x = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 9

Evaluate the second derivative at $x = \frac{\sqrt[4]{27}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \frac{{\sqrt[4]{27}}^{3}}{3^{3}} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.2

Combine $4$ and $\frac{{\sqrt[4]{27}}^{3}}{3^{3}}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.3

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 10.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4.2

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

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4.4

Cancel the common factor of $4$ .

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Step 10.1.4.4.1

Cancel the common factor.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4.4.2

Rewrite the expression.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{1}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.4.5

Evaluate the exponent.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.5

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

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 (\frac{27}{81}) - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.6

Cancel the common factor of $3$ .

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Step 10.1.6.1

Factor $3$ out of $81$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3 (27)} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.6.2

Cancel the common factor.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3 \cdot 27} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.6.3

Rewrite the expression.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (\frac{27}{27} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.7

Divide $27$ by $27$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (1 - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.8

Subtract $5$ from $1$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9

Simplify the numerator.

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Step 10.1.9.1

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

$\frac{\frac{4 \sqrt[4]{27^{3}}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9.2

Raise $27$ to the power of $3$ .

$\frac{\frac{4 \sqrt[4]{19683}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9.3

Rewrite $19683$ as $9^{4} \cdot 3$ .

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Step 10.1.9.3.1

Factor $6561$ out of $19683$ .

$\frac{\frac{4 \sqrt[4]{6561 (3)}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9.3.2

Rewrite $6561$ as $9^{4}$ .

$\frac{\frac{4 \sqrt[4]{9^{4} \cdot 3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9.4

Pull terms out from under the radical.

$\frac{\frac{4 \cdot 9 \sqrt[4]{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.9.5

Multiply $4$ by $9$ .

$\frac{\frac{36 \sqrt[4]{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.10

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

$\frac{\frac{36 \sqrt[4]{3}}{27} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.11

Cancel the common factor of $36$ and $27$ .

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Step 10.1.11.1

Factor $9$ out of $36 \sqrt[4]{3}$ .

$\frac{\frac{9 (4 \sqrt[4]{3})}{27} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.11.2

Cancel the common factors.

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Step 10.1.11.2.1

Factor $9$ out of $27$ .

$\frac{\frac{9 (4 \sqrt[4]{3})}{9 (3)} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.11.2.2

Cancel the common factor.

$\frac{\frac{9 (4 \sqrt[4]{3})}{9 \cdot 3} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.1.11.2.3

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 10.2.2

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 10.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} + 1)}^{3}}$

Step 10.2.2.2

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

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} + 1)}^{3}}$

Step 10.2.2.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 10.2.2.4

Cancel the common factor of $4$ .

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Step 10.2.2.4.1

Cancel the common factor.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 10.2.2.4.2

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{1}}{3^{4}} + 1)}^{3}}$

Step 10.2.2.5

Evaluate the exponent.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27}{3^{4}} + 1)}^{3}}$

Step 10.2.3

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

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27}{81} + 1)}^{3}}$

Step 10.2.4

Cancel the common factor of $27$ and $81$ .

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Step 10.2.4.1

Factor $27$ out of $27$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 (1)}{81} + 1)}^{3}}$

Step 10.2.4.2

Cancel the common factors.

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Step 10.2.4.2.1

Factor $27$ out of $81$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 10.2.4.2.2

Cancel the common factor.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 10.2.4.2.3

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1}{3} + 1)}^{3}}$

Step 10.2.5

Write $1$ as a fraction with a common denominator.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1}{3} + \frac{3}{3})}^{3}}$

Step 10.2.6

Combine the numerators over the common denominator.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1 + 3}{3})}^{3}}$

Step 10.2.7

Add $1$ and $3$ .

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

Step 10.2.8

Apply the product rule to $\frac{4}{3}$ .

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

Step 10.2.9

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

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{\frac{64}{3^{3}}}$

Step 10.2.10

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

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{\frac{64}{27}}$

Step 10.3

Combine fractions.

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Step 10.3.1

Combine $\frac{4 \sqrt[4]{3}}{3}$ and $- 4$ .

$\frac{\frac{4 \sqrt[4]{3} \cdot - 4}{3}}{\frac{64}{27}}$

Step 10.3.2

Simplify the expression.

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Step 10.3.2.1

Multiply $- 4$ by $4$ .

$\frac{\frac{- 16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 10.3.2.2

Move the negative in front of the fraction.

$\frac{- \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 10.5

Cancel the common factor of $16$ .

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Step 10.5.1

Move the leading negative in $- \frac{16 \sqrt[4]{3}}{3}$ into the numerator.

$\frac{- 16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 10.5.2

Factor $16$ out of $- 16 \sqrt[4]{3}$ .

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{64}$

Step 10.5.3

Factor $16$ out of $64$ .

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{16 (4)}$

Step 10.5.4

Cancel the common factor.

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{16 \cdot 4}$

Step 10.5.5

Rewrite the expression.

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

Step 10.6

Cancel the common factor of $3$ .

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Step 10.6.1

Factor $3$ out of $27$ .

$\frac{- \sqrt[4]{3}}{3} \cdot \frac{3 (9)}{4}$

Step 10.6.2

Cancel the common factor.

$\frac{- \sqrt[4]{3}}{3} \cdot \frac{3 \cdot 9}{4}$

Step 10.6.3

Rewrite the expression.

$- \sqrt[4]{3} \frac{9}{4}$

Step 10.7

Combine $\frac{9}{4}$ and $\sqrt[4]{3}$ .

$- \frac{9 \sqrt[4]{3}}{4}$

Step 11

$x = \frac{\sqrt[4]{27}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{\sqrt[4]{27}}{3}$ is a local maximum

Step 12

Find the y-value when $x = \frac{\sqrt[4]{27}}{3}$ .

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Step 12.1

Replace the variable $x$ with $\frac{\sqrt[4]{27}}{3}$ in the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\frac{\sqrt[4]{27}}{3}}{1 + {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{\sqrt[4]{27}}^{4}}{3^{4}}}$

Step 12.2.2.2

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 12.2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}}}$

Step 12.2.2.2.2

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

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}}}$

Step 12.2.2.2.3

Combine $\frac{1}{4}$ and $4$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 12.2.2.2.4

Cancel the common factor of $4$ .

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Step 12.2.2.2.4.1

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 12.2.2.2.4.2

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 12.2.2.2.5

Evaluate the exponent.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 12.2.2.3

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

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{81}}$

Step 12.2.2.4

Cancel the common factor of $27$ and $81$ .

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Step 12.2.2.4.1

Factor $27$ out of $27$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 (1)}{81}}$

Step 12.2.2.4.2

Cancel the common factors.

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Step 12.2.2.4.2.1

Factor $27$ out of $81$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 12.2.2.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 12.2.2.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{1}{3}}$

Step 12.2.2.5

Write $1$ as a fraction with a common denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3}{3} + \frac{1}{3}}$

Step 12.2.2.6

Combine the numerators over the common denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3 + 1}{3}}$

Step 12.2.2.7

Add $3$ and $1$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{4}{3}}$

Step 12.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot (1 (\frac{3}{4}))$

Step 12.2.4

Multiply $\frac{3}{4}$ by $1$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 12.2.5

Cancel the common factor of $3$ .

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Step 12.2.5.1

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 12.2.5.2

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \sqrt[4]{27} (\frac{1}{4})$

Step 12.2.6

Combine $\sqrt[4]{27}$ and $\frac{1}{4}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{4}$

Step 12.2.7

The final answer is $\frac{\sqrt[4]{27}}{4}$ .

$y = \frac{\sqrt[4]{27}}{4}$

Step 13

Evaluate the second derivative at $x = - \frac{\sqrt[4]{27}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 {(- \frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{4 {(- 1)}^{3} {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.2

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

$\frac{4 \cdot - 1 {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.3

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{{\sqrt[4]{27}}^{3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.4

Simplify the numerator.

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Step 14.1.4.1

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{27^{3}}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.4.2

Raise $27$ to the power of $3$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{19683}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.4.3

Rewrite $19683$ as $9^{4} \cdot 3$ .

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Step 14.1.4.3.1

Factor $6561$ out of $19683$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{6561 (3)}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.4.3.2

Rewrite $6561$ as $9^{4}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{9^{4} \cdot 3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.4.4

Pull terms out from under the radical.

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.5

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

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{27} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.6

Cancel the common factor of $9$ and $27$ .

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Step 14.1.6.1

Factor $9$ out of $9 \sqrt[4]{3}$ .

$\frac{4 \cdot - 1 \frac{9 (\sqrt[4]{3})}{27} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.6.2

Cancel the common factors.

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Step 14.1.6.2.1

Factor $9$ out of $27$ .

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{9 \cdot 3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.6.2.2

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{9 \cdot 3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.6.2.3

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7

Simplify each term.

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Step 14.1.7.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 14.1.7.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 ({(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 ({(- 1)}^{4} \frac{{\sqrt[4]{27}}^{4}}{3^{4}}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.2

Raise $- 1$ to the power of $4$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 (1 \frac{{\sqrt[4]{27}}^{4}}{3^{4}}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 14.1.7.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4.2

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

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4.4

Cancel the common factor of $4$ .

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Step 14.1.7.4.4.1

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4.4.2

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{1}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.4.5

Evaluate the exponent.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.5

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

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 (\frac{27}{81}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.6

Cancel the common factor of $3$ .

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Step 14.1.7.6.1

Factor $3$ out of $81$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3 (27)} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.6.2

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3 \cdot 27} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.6.3

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (\frac{27}{27} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.7.7

Divide $27$ by $27$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (1 - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.8

Subtract $5$ from $1$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} \cdot - 4}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.9

Combine exponents.

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Step 14.1.9.1

Factor out negative.

$\frac{- (4 \frac{\sqrt[4]{3}}{3} \cdot - 4)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.9.2

Combine $4$ and $\frac{\sqrt[4]{3}}{3}$ .

$\frac{- (\frac{4 \sqrt[4]{3}}{3} \cdot - 4)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.9.3

Combine $\frac{4 \sqrt[4]{3}}{3}$ and $- 4$ .

$\frac{- \frac{4 \sqrt[4]{3} \cdot - 4}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.9.4

Multiply $- 4$ by $4$ .

$\frac{- \frac{- 16 \sqrt[4]{3}}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.1.10

Move the negative in front of the fraction.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 14.2.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 14.2.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- 1)}^{4} \frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 14.2.2

Raise $- 1$ to the power of $4$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(1 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 14.2.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 14.2.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 14.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} + 1)}^{3}}$

Step 14.2.4.2

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

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} + 1)}^{3}}$

Step 14.2.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 14.2.4.4

Cancel the common factor of $4$ .

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Step 14.2.4.4.1

Cancel the common factor.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 14.2.4.4.2

Rewrite the expression.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{1}}{3^{4}} + 1)}^{3}}$

Step 14.2.4.5

Evaluate the exponent.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27}{3^{4}} + 1)}^{3}}$

Step 14.2.5

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

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27}{81} + 1)}^{3}}$

Step 14.2.6

Cancel the common factor of $27$ and $81$ .

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Step 14.2.6.1

Factor $27$ out of $27$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 (1)}{81} + 1)}^{3}}$

Step 14.2.6.2

Cancel the common factors.

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Step 14.2.6.2.1

Factor $27$ out of $81$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 14.2.6.2.2

Cancel the common factor.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 14.2.6.2.3

Rewrite the expression.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1}{3} + 1)}^{3}}$

Step 14.2.7

Write $1$ as a fraction with a common denominator.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1}{3} + \frac{3}{3})}^{3}}$

Step 14.2.8

Combine the numerators over the common denominator.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1 + 3}{3})}^{3}}$

Step 14.2.9

Add $1$ and $3$ .

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

Step 14.2.10

Apply the product rule to $\frac{4}{3}$ .

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

Step 14.2.11

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

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{3^{3}}}$

Step 14.2.12

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

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1 \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 14.3.2

Multiply $\frac{16 \sqrt[4]{3}}{3}$ by $1$ .

$\frac{\frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 14.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 14.5

Cancel the common factor of $16$ .

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Step 14.5.1

Factor $16$ out of $16 \sqrt[4]{3}$ .

$\frac{16 (\sqrt[4]{3})}{3} \cdot \frac{27}{64}$

Step 14.5.2

Factor $16$ out of $64$ .

$\frac{16 (\sqrt[4]{3})}{3} \cdot \frac{27}{16 (4)}$

Step 14.5.3

Cancel the common factor.

$\frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{16 \cdot 4}$

Step 14.5.4

Rewrite the expression.

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

Step 14.6

Cancel the common factor of $3$ .

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Step 14.6.1

Factor $3$ out of $27$ .

$\frac{\sqrt[4]{3}}{3} \cdot \frac{3 (9)}{4}$

Step 14.6.2

Cancel the common factor.

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

Step 14.6.3

Rewrite the expression.

$\sqrt[4]{3} \frac{9}{4}$

Step 14.7

Combine $\sqrt[4]{3}$ and $\frac{9}{4}$ .

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

Step 14.8

Move $9$ to the left of $\sqrt[4]{3}$ .

$\frac{9 \sqrt[4]{3}}{4}$

Step 15

$x = - \frac{\sqrt[4]{27}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt[4]{27}}{3}$ is a local minimum

Step 16

Find the y-value when $x = - \frac{\sqrt[4]{27}}{3}$ .

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Step 16.1

Replace the variable $x$ with $- \frac{\sqrt[4]{27}}{3}$ in the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \frac{\sqrt[4]{27}}{3}}{1 + {(- \frac{\sqrt[4]{27}}{3})}^{4}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- \frac{\sqrt[4]{27}}{3})}^{4}}$

Step 16.2.2

Simplify the denominator.

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Step 16.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 16.2.2.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 16.2.2.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- 1)}^{4} (\frac{{\sqrt[4]{27}}^{4}}{3^{4}})}$

Step 16.2.2.2

Raise $- 1$ to the power of $4$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + 1 (\frac{{\sqrt[4]{27}}^{4}}{3^{4}})}$

Step 16.2.2.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{\sqrt[4]{27}}^{4}}{3^{4}}}$

Step 16.2.2.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

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Step 16.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}}}$

Step 16.2.2.4.2

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

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}}}$

Step 16.2.2.4.3

Combine $\frac{1}{4}$ and $4$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 16.2.2.4.4

Cancel the common factor of $4$ .

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Step 16.2.2.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 16.2.2.4.4.2

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 16.2.2.4.5

Evaluate the exponent.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 16.2.2.5

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

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{81}}$

Step 16.2.2.6

Cancel the common factor of $27$ and $81$ .

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Step 16.2.2.6.1

Factor $27$ out of $27$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 (1)}{81}}$

Step 16.2.2.6.2

Cancel the common factors.

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Step 16.2.2.6.2.1

Factor $27$ out of $81$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 16.2.2.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 16.2.2.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{1}{3}}$

Step 16.2.2.7

Write $1$ as a fraction with a common denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3}{3} + \frac{1}{3}}$

Step 16.2.2.8

Combine the numerators over the common denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3 + 1}{3}}$

Step 16.2.2.9

Add $3$ and $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{4}{3}}$

Step 16.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot (1 (\frac{3}{4}))$

Step 16.2.4

Multiply $\frac{3}{4}$ by $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 16.2.5

Cancel the common factor of $3$ .

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Step 16.2.5.1

Move the leading negative in $- \frac{\sqrt[4]{27}}{3}$ into the numerator.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 16.2.5.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 16.2.5.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \sqrt[4]{27} \frac{1}{4}$

Step 16.2.6

Combine $\frac{1}{4}$ and $\sqrt[4]{27}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{4}$

Step 16.2.7

The final answer is $- \frac{\sqrt[4]{27}}{4}$ .

$y = - \frac{\sqrt[4]{27}}{4}$

Step 17

These are the local extrema for $f (x) = \frac{x}{1 + x^{4}}$ .

$(\frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{4})$ is a local maxima

$(- \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{4})$ is a local minima

Step 18