Calculus Examples

$f (x) = \frac{7 x^{2} + 28}{x^{4} - 16}$

Step 1

Find the first derivative of the function.

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Step 1.1

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 $f (x) = 7 x^{2} + 28$ and $g (x) = x^{4} - 16$ .

$\frac{(x^{4} - 16) \frac{d}{d x} [7 x^{2} + 28] - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $7 x^{2} + 28$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [28]$ .

$\frac{(x^{4} - 16) (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\frac{(x^{4} - 16) (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.3

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

$\frac{(x^{4} - 16) (7 (2 x) + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.4

Multiply $2$ by $7$ .

$\frac{(x^{4} - 16) (14 x + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.5

Since $28$ is constant with respect to $x$ , the derivative of $28$ with respect to $x$ is $0$ .

$\frac{(x^{4} - 16) (14 x + 0) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $14 x$ and $0$ .

$\frac{(x^{4} - 16) (14 x) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.6.2

Move $14$ to the left of $x^{4} - 16$ .

$\frac{14 \cdot (x^{4} - 16) x - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 1.2.7

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16])}{{(x^{4} - 16)}^{2}}$

Step 1.2.8

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

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3} + \frac{d}{d x} [- 16])}{{(x^{4} - 16)}^{2}}$

Step 1.2.9

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3} + 0)}{{(x^{4} - 16)}^{2}}$

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

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

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3})}{{(x^{4} - 16)}^{2}}$

Step 1.2.10.2

Multiply $4$ by $- 1$ .

$\frac{14 (x^{4} - 16) x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(14 x^{4} + 14 \cdot - 16) x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.3

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x + (- 4 (7 x^{2}) - 4 \cdot 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.4

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

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

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Step 1.3.5.1.1.1

Move $x$ .

$\frac{14 (x \cdot x^{4}) + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.1.2

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

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Step 1.3.5.1.1.2.1

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

$\frac{14 (x^{1} x^{4}) + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.1.2.2

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

$\frac{14 x^{1 + 4} + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.1.3

Add $1$ and $4$ .

$\frac{14 x^{5} + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.2

Multiply $14$ by $- 16$ .

$\frac{14 x^{5} - 224 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.3

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.3.5.1.3.1

Move $x^{3}$ .

$\frac{14 x^{5} - 224 x - 4 (7 (x^{3} x^{2})) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.3.2

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

$\frac{14 x^{5} - 224 x - 4 (7 x^{3 + 2}) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.3.3

Add $3$ and $2$ .

$\frac{14 x^{5} - 224 x - 4 (7 x^{5}) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.4

Multiply $7$ by $- 4$ .

$\frac{14 x^{5} - 224 x - 28 x^{5} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.1.5

Multiply $- 4$ by $28$ .

$\frac{14 x^{5} - 224 x - 28 x^{5} - 112 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.5.2

Subtract $28 x^{5}$ from $14 x^{5}$ .

$\frac{- 14 x^{5} - 224 x - 112 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 1.3.6

Reorder terms.

$\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

$f'' (x) = \frac{{(x^{4} - 16)}^{2} \frac{d}{d x} (- 14 x^{5} - 112 x^{3} - 224 x) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{({(x^{4} - 16)}^{2})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${({(x^{4} - 16)}^{2})}^{2}$ .

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Step 2.2.1.1

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

$f'' (x) = \frac{{(x^{4} - 16)}^{2} \frac{d}{d x} (- 14 x^{5} - 112 x^{3} - 224 x) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{2 \cdot 2}}$

Step 2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} \frac{d}{d x} (- 14 x^{5} - 112 x^{3} - 224 x) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of $- 14 x^{5} - 112 x^{3} - 224 x$ with respect to $x$ is $\frac{d}{d x} [- 14 x^{5}] + \frac{d}{d x} [- 112 x^{3}] + \frac{d}{d x} [- 224 x]$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (\frac{d}{d x} (- 14 x^{5}) + \frac{d}{d x} (- 112 x^{3}) + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.3

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x^{5}$ with respect to $x$ is $- 14 \frac{d}{d x} [x^{5}]$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 14 \frac{d}{d x} x^{5} + \frac{d}{d x} (- 112 x^{3}) + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 14 (5 x^{4}) + \frac{d}{d x} (- 112 x^{3}) + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.5

Multiply $5$ by $- 14$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} + \frac{d}{d x} (- 112 x^{3}) + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.6

Since $- 112$ is constant with respect to $x$ , the derivative of $- 112 x^{3}$ with respect to $x$ is $- 112 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 112 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.7

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

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 112 (3 x^{2}) + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.8

Multiply $3$ by $- 112$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} + \frac{d}{d x} (- 224 x)) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.9

Since $- 224$ is constant with respect to $x$ , the derivative of $- 224 x$ with respect to $x$ is $- 224 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224 \frac{d}{d x} x) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.10

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

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224 \cdot 1) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.2.11

Multiply $- 224$ by $1$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - (- 14 x^{5} - 112 x^{3} - 224 x) \frac{d}{d x} {(x^{4} - 16)}^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{4} - 16$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{4} - 16$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.3.2

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

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - (- 14 x^{5} - 112 x^{3} - 224 x) (2 u \frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.3.3

Replace all occurrences of $u$ with $x^{4} - 16$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - (- 14 x^{5} - 112 x^{3} - 224 x) (2 (x^{4} - 16) \frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) ((x^{4} - 16) \frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.4.2

Factor $x^{4} - 16$ out of ${(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) ((x^{4} - 16) \frac{d}{d x} [x^{4} - 16])$ .

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Step 2.4.2.1

Factor $x^{4} - 16$ out of ${(x^{4} - 16)}^{2} (- 70 x^{4} - 336 x^{2} - 224)$ .

$f'' (x) = \frac{(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224)) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) ((x^{4} - 16) \frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.4.2.2

Factor $x^{4} - 16$ out of $- 2 (- 14 x^{5} - 112 x^{3} - 224 x) ((x^{4} - 16) \frac{d}{d x} [x^{4} - 16])$ .

$f'' (x) = \frac{(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224)) + (x^{4} - 16) (- 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4} - 16)))}{{(x^{4} - 16)}^{4}}$

Step 2.4.2.3

Factor $x^{4} - 16$ out of $(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224)) + (x^{4} - 16) (- 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} [x^{4} - 16]))$ .

$f'' (x) = \frac{(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4} - 16)))}{{(x^{4} - 16)}^{4}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

Factor $x^{4} - 16$ out of ${(x^{4} - 16)}^{4}$ .

$f'' (x) = \frac{(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4} - 16)))}{(x^{4} - 16) {(x^{4} - 16)}^{3}}$

Step 2.5.2

Cancel the common factor.

$f'' (x) = \frac{(x^{4} - 16) ((x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4} - 16)))}{(x^{4} - 16) {(x^{4} - 16)}^{3}}$

Step 2.5.3

Rewrite the expression.

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4} - 16))}{{(x^{4} - 16)}^{3}}$

Step 2.6

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 16))}{{(x^{4} - 16)}^{3}}$

Step 2.7

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

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (4 x^{3} + \frac{d}{d x} (- 16))}{{(x^{4} - 16)}^{3}}$

Step 2.8

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (4 x^{3} + 0)}{{(x^{4} - 16)}^{3}}$

Step 2.9

Simplify the expression.

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Step 2.9.1

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

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 2 (- 14 x^{5} - 112 x^{3} - 224 x) (4 x^{3})}{{(x^{4} - 16)}^{3}}$

Step 2.9.2

Multiply $4$ by $- 2$ .

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 8 (- 14 x^{5} - 112 x^{3} - 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) + (- 8 (- 14 x^{5}) - 8 (- 112 x^{3}) - 8 (- 224 x)) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.2

Apply the distributive property.

$f'' (x) = \frac{(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224) - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3

Simplify the numerator.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

Expand $(x^{4} - 16) (- 70 x^{4} - 336 x^{2} - 224)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{x^{4} (- 70 x^{4}) + x^{4} (- 336 x^{2}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2

Simplify each term.

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Step 2.10.3.1.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 70 x^{4} x^{4} + x^{4} (- 336 x^{2}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.2

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

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Step 2.10.3.1.2.2.1

Move $x^{4}$ .

$f'' (x) = \frac{- 70 (x^{4} x^{4}) + x^{4} (- 336 x^{2}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.2.2

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

$f'' (x) = \frac{- 70 x^{4 + 4} + x^{4} (- 336 x^{2}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.2.3

Add $4$ and $4$ .

$f'' (x) = \frac{- 70 x^{8} + x^{4} (- 336 x^{2}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 70 x^{8} - 336 x^{4} x^{2} + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.4

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.10.3.1.2.4.1

Move $x^{2}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 (x^{2} x^{4}) + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.4.2

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

$f'' (x) = \frac{- 70 x^{8} - 336 x^{2 + 4} + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.4.3

Add $2$ and $4$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + x^{4} \cdot - 224 - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.5

Move $- 224$ to the left of $x^{4}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} - 224 \cdot x^{4} - 16 (- 70 x^{4}) - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.6

Multiply $- 70$ by $- 16$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} - 224 x^{4} + 1120 x^{4} - 16 (- 336 x^{2}) - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.7

Multiply $- 336$ by $- 16$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} - 224 x^{4} + 1120 x^{4} + 5376 x^{2} - 16 \cdot - 224 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.2.8

Multiply $- 16$ by $- 224$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} - 224 x^{4} + 1120 x^{4} + 5376 x^{2} + 3584 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.3

Add $- 224 x^{4}$ and $1120 x^{4}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 - 8 (- 14 x^{5}) x^{3} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.4

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 2.10.3.1.4.1

Move $x^{3}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 - 8 (- 14 (x^{3} x^{5})) - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.4.2

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

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 - 8 (- 14 x^{3 + 5}) - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.4.3

Add $3$ and $5$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 - 8 (- 14 x^{8}) - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.5

Multiply $- 14$ by $- 8$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} - 8 (- 112 x^{3}) x^{3} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.6

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

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Step 2.10.3.1.6.1

Move $x^{3}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} - 8 (- 112 (x^{3} x^{3})) - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.6.2

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

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} - 8 (- 112 x^{3 + 3}) - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.6.3

Add $3$ and $3$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} - 8 (- 112 x^{6}) - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.7

Multiply $- 112$ by $- 8$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} - 8 (- 224 x) x^{3}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.8

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

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Step 2.10.3.1.8.1

Move $x^{3}$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} - 8 (- 224 (x^{3} x))}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.8.2

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

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Step 2.10.3.1.8.2.1

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

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} - 8 (- 224 (x^{3} x))}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.8.2.2

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

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} - 8 (- 224 x^{3 + 1})}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.8.3

Add $3$ and $1$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} - 8 (- 224 x^{4})}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.1.9

Multiply $- 224$ by $- 8$ .

$f'' (x) = \frac{- 70 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 112 x^{8} + 896 x^{6} + 1792 x^{4}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.2

Add $- 70 x^{8}$ and $112 x^{8}$ .

$f'' (x) = \frac{42 x^{8} - 336 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 896 x^{6} + 1792 x^{4}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.3

Add $- 336 x^{6}$ and $896 x^{6}$ .

$f'' (x) = \frac{42 x^{8} + 560 x^{6} + 896 x^{4} + 5376 x^{2} + 3584 + 1792 x^{4}}{{(x^{4} - 16)}^{3}}$

Step 2.10.3.4

Add $896 x^{4}$ and $1792 x^{4}$ .

$f'' (x) = \frac{42 x^{8} + 560 x^{6} + 2688 x^{4} + 5376 x^{2} + 3584}{{(x^{4} - 16)}^{3}}$

Step 2.10.4

Factor $14$ out of $42 x^{8} + 560 x^{6} + 2688 x^{4} + 5376 x^{2} + 3584$ .

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Step 2.10.4.1

Factor $14$ out of $42 x^{8}$ .

$f'' (x) = \frac{14 (3 x^{8}) + 560 x^{6} + 2688 x^{4} + 5376 x^{2} + 3584}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.2

Factor $14$ out of $560 x^{6}$ .

$f'' (x) = \frac{14 (3 x^{8}) + 14 (40 x^{6}) + 2688 x^{4} + 5376 x^{2} + 3584}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.3

Factor $14$ out of $2688 x^{4}$ .

$f'' (x) = \frac{14 (3 x^{8}) + 14 (40 x^{6}) + 14 (192 x^{4}) + 5376 x^{2} + 3584}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.4

Factor $14$ out of $5376 x^{2}$ .

$f'' (x) = \frac{14 (3 x^{8}) + 14 (40 x^{6}) + 14 (192 x^{4}) + 14 (384 x^{2}) + 3584}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.5

Factor $14$ out of $3584$ .

$f'' (x) = \frac{14 (3 x^{8}) + 14 (40 x^{6}) + 14 (192 x^{4}) + 14 (384 x^{2}) + 14 (256)}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.6

Factor $14$ out of $14 (3 x^{8}) + 14 (40 x^{6})$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6}) + 14 (192 x^{4}) + 14 (384 x^{2}) + 14 (256)}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.7

Factor $14$ out of $14 (3 x^{8} + 40 x^{6}) + 14 (192 x^{4})$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4}) + 14 (384 x^{2}) + 14 (256)}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.8

Factor $14$ out of $14 (3 x^{8} + 40 x^{6} + 192 x^{4}) + 14 (384 x^{2})$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2}) + 14 (256)}{{(x^{4} - 16)}^{3}}$

Step 2.10.4.9

Factor $14$ out of $14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2}) + 14 (256)$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{4} - 16)}^{3}}$

Step 2.10.5

Simplify the denominator.

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Step 2.10.5.1

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{({(x^{2})}^{2} - 16)}^{3}}$

Step 2.10.5.2

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{({(x^{2})}^{2} - 4^{2})}^{3}}$

Step 2.10.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 4$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x^{2} + 4) (x^{2} - 4))}^{3}}$

Step 2.10.5.4

Simplify.

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Step 2.10.5.4.1

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x^{2} + 4) (x^{2} - 2^{2}))}^{3}}$

Step 2.10.5.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x^{2} + 4) (x + 2) (x - 2))}^{3}}$

Step 2.10.5.5

Apply the product rule to $(x^{2} + 4) (x + 2) (x - 2)$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x^{2} + 4) (x + 2))}^{3} {(x - 2)}^{3}}$

Step 2.10.5.6

Expand $(x^{2} + 4) (x + 2)$ using the FOIL Method.

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Step 2.10.5.6.1

Apply the distributive property.

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2} (x + 2) + 4 (x + 2))}^{3} {(x - 2)}^{3}}$

Step 2.10.5.6.2

Apply the distributive property.

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2} x + x^{2} \cdot 2 + 4 (x + 2))}^{3} {(x - 2)}^{3}}$

Step 2.10.5.6.3

Apply the distributive property.

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2} x + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.7

Simplify each term.

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Step 2.10.5.7.1

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

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Step 2.10.5.7.1.1

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

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Step 2.10.5.7.1.1.1

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2} x + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.7.1.1.2

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2 + 1} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.7.1.2

Add $2$ and $1$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{3} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.7.2

Move $2$ to the left of $x^{2}$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{3} + 2 \cdot x^{2} + 4 x + 4 \cdot 2)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.7.3

Multiply $4$ by $2$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{3} + 2 x^{2} + 4 x + 8)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.8

Factor out the greatest common factor from each group.

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Step 2.10.5.8.1

Group the first two terms and the last two terms.

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x^{3} + 2 x^{2}) + 4 x + 8)}^{3} {(x - 2)}^{3}}$

Step 2.10.5.8.2

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

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x^{2} (x + 2) + 4 (x + 2))}^{3} {(x - 2)}^{3}}$

Step 2.10.5.9

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{((x + 2) (x^{2} + 4))}^{3} {(x - 2)}^{3}}$

Step 2.10.5.10

Apply the product rule to $(x + 2) (x^{2} + 4)$ .

$f'' (x) = \frac{14 (3 x^{8} + 40 x^{6} + 192 x^{4} + 384 x^{2} + 256)}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}}$

Step 3

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

$\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}} = 0$

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

$\frac{(x^{4} - 16) \frac{d}{d x} [7 x^{2} + 28] - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $7 x^{2} + 28$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [28]$ .

$\frac{(x^{4} - 16) (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\frac{(x^{4} - 16) (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

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 $n = 2$ .

$\frac{(x^{4} - 16) (7 (2 x) + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.4

Multiply $2$ by $7$ .

$\frac{(x^{4} - 16) (14 x + \frac{d}{d x} [28]) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.5

Since $28$ is constant with respect to $x$ , the derivative of $28$ with respect to $x$ is $0$ .

$\frac{(x^{4} - 16) (14 x + 0) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.6

Simplify the expression.

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Step 4.1.2.6.1

Add $14 x$ and $0$ .

$\frac{(x^{4} - 16) (14 x) - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.6.2

Move $14$ to the left of $x^{4} - 16$ .

$\frac{14 \cdot (x^{4} - 16) x - (7 x^{2} + 28) \frac{d}{d x} [x^{4} - 16]}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.7

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16])}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.8

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

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3} + \frac{d}{d x} [- 16])}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.9

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3} + 0)}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.10

Simplify the expression.

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Step 4.1.2.10.1

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

$\frac{14 (x^{4} - 16) x - (7 x^{2} + 28) (4 x^{3})}{{(x^{4} - 16)}^{2}}$

Step 4.1.2.10.2

Multiply $4$ by $- 1$ .

$\frac{14 (x^{4} - 16) x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$\frac{(14 x^{4} + 14 \cdot - 16) x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.2

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x - 4 (7 x^{2} + 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.3

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x + (- 4 (7 x^{2}) - 4 \cdot 28) x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.4

Apply the distributive property.

$\frac{14 x^{4} x + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5

Simplify the numerator.

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Step 4.1.3.5.1

Simplify each term.

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Step 4.1.3.5.1.1

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

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Step 4.1.3.5.1.1.1

Move $x$ .

$\frac{14 (x \cdot x^{4}) + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.1.2

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

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Step 4.1.3.5.1.1.2.1

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

$\frac{14 (x^{1} x^{4}) + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.1.2.2

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

$\frac{14 x^{1 + 4} + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.1.3

Add $1$ and $4$ .

$\frac{14 x^{5} + 14 \cdot - 16 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.2

Multiply $14$ by $- 16$ .

$\frac{14 x^{5} - 224 x - 4 (7 x^{2}) x^{3} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.3

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 4.1.3.5.1.3.1

Move $x^{3}$ .

$\frac{14 x^{5} - 224 x - 4 (7 (x^{3} x^{2})) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.3.2

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

$\frac{14 x^{5} - 224 x - 4 (7 x^{3 + 2}) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.3.3

Add $3$ and $2$ .

$\frac{14 x^{5} - 224 x - 4 (7 x^{5}) - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.4

Multiply $7$ by $- 4$ .

$\frac{14 x^{5} - 224 x - 28 x^{5} - 4 \cdot 28 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.1.5

Multiply $- 4$ by $28$ .

$\frac{14 x^{5} - 224 x - 28 x^{5} - 112 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.5.2

Subtract $28 x^{5}$ from $14 x^{5}$ .

$\frac{- 14 x^{5} - 224 x - 112 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 4.1.3.6

Reorder terms.

$f' (x) = \frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}}$ .

$\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

$- 14 x^{5} - 112 x^{3} - 224 x = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Factor the left side of the equation.

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Step 5.3.1.1

Factor $- 14 x$ out of $- 14 x^{5} - 112 x^{3} - 224 x$ .

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Step 5.3.1.1.1

Factor $- 14 x$ out of $- 14 x^{5}$ .

$- 14 x \cdot x^{4} - 112 x^{3} - 224 x = 0$

Step 5.3.1.1.2

Factor $- 14 x$ out of $- 112 x^{3}$ .

$- 14 x \cdot x^{4} - 14 x (8 x^{2}) - 224 x = 0$

Step 5.3.1.1.3

Factor $- 14 x$ out of $- 224 x$ .

$- 14 x \cdot x^{4} - 14 x (8 x^{2}) - 14 x \cdot 16 = 0$

Step 5.3.1.1.4

Factor $- 14 x$ out of $- 14 x (x^{4}) - 14 x (8 x^{2})$ .

$- 14 x (x^{4} + 8 x^{2}) - 14 x \cdot 16 = 0$

Step 5.3.1.1.5

Factor $- 14 x$ out of $- 14 x (x^{4} + 8 x^{2}) - 14 x (16)$ .

$- 14 x (x^{4} + 8 x^{2} + 16) = 0$

Step 5.3.1.2

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

$- 14 x ({(x^{2})}^{2} + 8 x^{2} + 16) = 0$

Step 5.3.1.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 14 x (u^{2} + 8 u + 16) = 0$

Step 5.3.1.4

Factor using the perfect square rule.

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Step 5.3.1.4.1

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

$- 14 x (u^{2} + 8 u + 4^{2}) = 0$

Step 5.3.1.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Step 5.3.1.4.3

Rewrite the polynomial.

$- 14 x (u^{2} + 2 \cdot u \cdot 4 + 4^{2}) = 0$

Step 5.3.1.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 4$ .

$- 14 x {(u + 4)}^{2} = 0$

Step 5.3.1.5

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

$- 14 x {(x^{2} + 4)}^{2} = 0$

Step 5.3.2

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

$x = 0$

${(x^{2} + 4)}^{2} = 0$

Step 5.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.4

Set ${(x^{2} + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 5.3.4.1

Set ${(x^{2} + 4)}^{2}$ equal to $0$ .

${(x^{2} + 4)}^{2} = 0$

Step 5.3.4.2

Solve ${(x^{2} + 4)}^{2} = 0$ for $x$ .

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Step 5.3.4.2.1

Set the $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 5.3.4.2.2

Solve for $x$ .

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Step 5.3.4.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 5.3.4.2.2.2

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

$x = \pm \sqrt{- 4}$

Step 5.3.4.2.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 5.3.4.2.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 5.3.4.2.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 5.3.4.2.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 5.3.4.2.2.3.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 5.3.4.2.2.3.5

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

$x = \pm i \cdot 2$

Step 5.3.4.2.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 5.3.4.2.2.4

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

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Step 5.3.4.2.2.4.1

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

$x = 2 i$

Step 5.3.4.2.2.4.2

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

$x = - 2 i$

Step 5.3.4.2.2.4.3

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

$x = 2 i, - 2 i$

Step 5.3.5

The final solution is all the values that make $- 14 x {(x^{2} + 4)}^{2} = 0$ true.

$x = 0, 2 i, - 2 i$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{- 14 x^{5} - 112 x^{3} - 224 x}{{(x^{4} - 16)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 6.2

Solve for $x$ .

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Step 6.2.1

Factor the left side of the equation.

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

${({(x^{2})}^{2} - 4^{2})}^{2} = 0$

Step 6.2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 4$ .

${((x^{2} + 4) (x^{2} - 4))}^{2} = 0$

Step 6.2.1.4

Simplify.

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Step 6.2.1.4.1

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

${((x^{2} + 4) (x^{2} - 2^{2}))}^{2} = 0$

Step 6.2.1.4.2

Factor.

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Step 6.2.1.4.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

${((x^{2} + 4) ((x + 2) (x - 2)))}^{2} = 0$

Step 6.2.1.4.2.2

Remove unnecessary parentheses.

${((x^{2} + 4) (x + 2) (x - 2))}^{2} = 0$

Step 6.2.1.5

Apply the product rule to $(x^{2} + 4) (x + 2) (x - 2)$ .

${((x^{2} + 4) (x + 2))}^{2} {(x - 2)}^{2} = 0$

Step 6.2.1.6

Apply the product rule to $(x^{2} + 4) (x + 2)$ .

${(x^{2} + 4)}^{2} {(x + 2)}^{2} {(x - 2)}^{2} = 0$

Step 6.2.2

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

${(x^{2} + 4)}^{2} = 0$

${(x + 2)}^{2} = 0$

${(x - 2)}^{2} = 0$

Step 6.2.3

Set ${(x^{2} + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.2.3.1

Set ${(x^{2} + 4)}^{2}$ equal to $0$ .

${(x^{2} + 4)}^{2} = 0$

Step 6.2.3.2

Solve ${(x^{2} + 4)}^{2} = 0$ for $x$ .

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Step 6.2.3.2.1

Set the $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 6.2.3.2.2

Solve for $x$ .

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Step 6.2.3.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 6.2.3.2.2.2

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

$x = \pm \sqrt{- 4}$

Step 6.2.3.2.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 6.2.3.2.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 6.2.3.2.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 6.2.3.2.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 6.2.3.2.2.3.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 6.2.3.2.2.3.5

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

$x = \pm i \cdot 2$

Step 6.2.3.2.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 6.2.3.2.2.4

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

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Step 6.2.3.2.2.4.1

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

$x = 2 i$

Step 6.2.3.2.2.4.2

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

$x = - 2 i$

Step 6.2.3.2.2.4.3

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

$x = 2 i, - 2 i$

Step 6.2.4

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.2.4.1

Set ${(x + 2)}^{2}$ equal to $0$ .

${(x + 2)}^{2} = 0$

Step 6.2.4.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 6.2.4.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.2.4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.2.5

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.2.5.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 6.2.5.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 6.2.5.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.2.5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.2.6

The final solution is all the values that make ${(x^{2} + 4)}^{2} {(x + 2)}^{2} {(x - 2)}^{2} = 0$ true.

$x = 2 i, - 2 i, - 2, 2$

Step 6.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 7

Critical points to evaluate.

$x = 0$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{14 (3 {(0)}^{8} + 40 {(0)}^{6} + 192 {(0)}^{4} + 384 {(0)}^{2} + 256)}{{((0) + 2)}^{3} {({(0)}^{2} + 4)}^{3} {((0) - 2)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{14 (3 \cdot 0 + 40 \cdot 0^{6} + 192 \cdot 0^{4} + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.2

Multiply $3$ by $0$ .

$\frac{14 (0 + 40 \cdot 0^{6} + 192 \cdot 0^{4} + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.3

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

$\frac{14 (0 + 40 \cdot 0 + 192 \cdot 0^{4} + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.4

Multiply $40$ by $0$ .

$\frac{14 (0 + 0 + 192 \cdot 0^{4} + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.5

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

$\frac{14 (0 + 0 + 192 \cdot 0 + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.6

Multiply $192$ by $0$ .

$\frac{14 (0 + 0 + 0 + 384 \cdot 0^{2} + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.7

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

$\frac{14 (0 + 0 + 0 + 384 \cdot 0 + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.8

Multiply $384$ by $0$ .

$\frac{14 (0 + 0 + 0 + 0 + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.9

Add $0$ and $0$ .

$\frac{14 (0 + 0 + 0 + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.10

Add $0$ and $0$ .

$\frac{14 (0 + 0 + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.11

Add $0$ and $0$ .

$\frac{14 (0 + 256)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.12

Add $0$ and $256$ .

$\frac{14 \cdot 256}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{14 \cdot 256}{{(- 1 (0) + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{14 \cdot 256}{{(- 1 \cdot 0 - 1 \cdot - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 2$ .

$\frac{14 \cdot 256}{{(- 1 \cdot (0 - 2))}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.4

Apply the product rule to $- 1 \cdot (0 - 2)$ .

$\frac{14 \cdot 256}{{(- 1)}^{3} \cdot {(0 - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.5

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

$\frac{14 \cdot 256}{- 1 \cdot {(0 - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.6

Multiply ${(0 - 2)}^{3}$ by ${(0 - 2)}^{3}$ by adding the exponents.

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Step 9.2.6.1

Move ${(0 - 2)}^{3}$ .

$\frac{14 \cdot 256}{- 1 \cdot ({(0 - 2)}^{3} {(0 - 2)}^{3}) {(0^{2} + 4)}^{3}}$

Step 9.2.6.2

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

$\frac{14 \cdot 256}{- 1 \cdot {(0 - 2)}^{3 + 3} {(0^{2} + 4)}^{3}}$

Step 9.2.6.3

Add $3$ and $3$ .

$\frac{14 \cdot 256}{- 1 \cdot {(0 - 2)}^{6} {(0^{2} + 4)}^{3}}$

Step 9.3

Multiply $14$ by $256$ .

$\frac{3584}{- 1 \cdot {(0 - 2)}^{6} {(0^{2} + 4)}^{3}}$

Step 9.4

Simplify the denominator.

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Step 9.4.1

Subtract $2$ from $0$ .

$\frac{3584}{- 1 \cdot {(- 2)}^{6} {(0^{2} + 4)}^{3}}$

Step 9.4.2

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

$\frac{3584}{- 1 \cdot {(- 2)}^{6} {(0 + 4)}^{3}}$

Step 9.4.3

Add $0$ and $4$ .

$\frac{3584}{- 1 \cdot {(- 2)}^{6} \cdot 4^{3}}$

Step 9.4.4

Combine exponents.

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Step 9.4.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{3584}{- 1 \cdot {(- 1 (2))}^{6} \cdot 4^{3}}$

Step 9.4.4.2

Apply the product rule to $- 1 (2)$ .

$\frac{3584}{- 1 \cdot ({(- 1)}^{6} \cdot 2^{6}) \cdot 4^{3}}$

Step 9.4.4.3

Raise $- 1$ to the power of $6$ .

$\frac{3584}{- 1 \cdot (1 \cdot 2^{6}) \cdot 4^{3}}$

Step 9.4.4.4

Multiply $2^{6}$ by $1$ .

$\frac{3584}{- 1 \cdot 2^{6} \cdot 4^{3}}$

Step 9.4.4.5

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

$\frac{3584}{- 1 \cdot ({(2^{2})}^{3} \cdot 2^{6})}$

Step 9.4.4.6

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

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Step 9.4.4.6.1

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

$\frac{3584}{- 1 \cdot (2^{2 \cdot 3} \cdot 2^{6})}$

Step 9.4.4.6.2

Multiply $2$ by $3$ .

$\frac{3584}{- 1 \cdot (2^{6} \cdot 2^{6})}$

Step 9.4.4.7

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

$\frac{3584}{- 1 \cdot 2^{6 + 6}}$

Step 9.4.4.8

Add $6$ and $6$ .

$\frac{3584}{- 1 \cdot 2^{12}}$

Step 9.4.5

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

$\frac{3584}{- 1 \cdot 4096}$

Step 9.5

Reduce the expression by cancelling the common factors.

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Step 9.5.1

Multiply $- 1$ by $4096$ .

$\frac{3584}{- 4096}$

Step 9.5.2

Cancel the common factor of $3584$ and $- 4096$ .

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Step 9.5.2.1

Factor $512$ out of $3584$ .

$\frac{512 (7)}{- 4096}$

Step 9.5.2.2

Cancel the common factors.

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Step 9.5.2.2.1

Factor $512$ out of $- 4096$ .

$\frac{512 \cdot 7}{512 \cdot - 8}$

Step 9.5.2.2.2

Cancel the common factor.

$\frac{512 \cdot 7}{512 \cdot - 8}$

Step 9.5.2.2.3

Rewrite the expression.

$\frac{7}{- 8}$

Step 9.5.3

Move the negative in front of the fraction.

$- \frac{7}{8}$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{7 {(0)}^{2} + 28}{{(0)}^{4} - 16}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the numerator.

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Step 11.2.1.1

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

$f (0) = \frac{7 \cdot 0 + 28}{0^{4} - 16}$

Step 11.2.1.2

Multiply $7$ by $0$ .

$f (0) = \frac{0 + 28}{0^{4} - 16}$

Step 11.2.1.3

Add $0$ and $28$ .

$f (0) = \frac{28}{0^{4} - 16}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

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

$f (0) = \frac{28}{0 - 16}$

Step 11.2.2.2

Subtract $16$ from $0$ .

$f (0) = \frac{28}{- 16}$

Step 11.2.3

Reduce the expression by cancelling the common factors.

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Step 11.2.3.1

Cancel the common factor of $28$ and $- 16$ .

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Step 11.2.3.1.1

Factor $4$ out of $28$ .

$f (0) = \frac{4 (7)}{- 16}$

Step 11.2.3.1.2

Cancel the common factors.

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Step 11.2.3.1.2.1

Factor $4$ out of $- 16$ .

$f (0) = \frac{4 \cdot 7}{4 \cdot - 4}$

Step 11.2.3.1.2.2

Cancel the common factor.

$f (0) = \frac{4 \cdot 7}{4 \cdot - 4}$

Step 11.2.3.1.2.3

Rewrite the expression.

$f (0) = \frac{7}{- 4}$

Step 11.2.3.2

Move the negative in front of the fraction.

$f (0) = - \frac{7}{4}$

Step 11.2.4

The final answer is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 12

These are the local extrema for $f (x) = \frac{7 x^{2} + 28}{x^{4} - 16}$ .

$(0, - \frac{7}{4})$ is a local maxima

Step 13