Calculus Examples

$\frac{x^{4}}{x^{4} - 256}$

Step 1

Write $\frac{x^{4}}{x^{4} - 256}$ as a function.

$f (x) = \frac{x^{4}}{x^{4} - 256}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $f (x) = x^{4}$ and $g (x) = x^{4} - 256$ .

$\frac{(x^{4} - 256) \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [x^{4} - 256]}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

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

$\frac{(x^{4} - 256) (4 x^{3}) - x^{4} \frac{d}{d x} [x^{4} - 256]}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2.2

Move $4$ to the left of $x^{4} - 256$ .

$\frac{4 \cdot (x^{4} - 256) x^{3} - x^{4} \frac{d}{d x} [x^{4} - 256]}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2.3

By the Sum Rule, the derivative of $x^{4} - 256$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 256]$ .

$\frac{4 (x^{4} - 256) x^{3} - x^{4} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 256])}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2.4

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

$\frac{4 (x^{4} - 256) x^{3} - x^{4} (4 x^{3} + \frac{d}{d x} [- 256])}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2.5

Since $- 256$ is constant with respect to $x$ , the derivative of $- 256$ with respect to $x$ is $0$ .

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

Step 2.1.1.2.6

Simplify the expression.

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Step 2.1.1.2.6.1

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

$\frac{4 (x^{4} - 256) x^{3} - x^{4} (4 x^{3})}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.2.6.2

Multiply $4$ by $- 1$ .

$\frac{4 (x^{4} - 256) x^{3} - 4 x^{4} x^{3}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.3

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

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Step 2.1.1.3.1

Move $x^{3}$ .

$\frac{4 (x^{4} - 256) x^{3} - 4 (x^{3} x^{4})}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.3.2

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

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

Step 2.1.1.3.3

Add $3$ and $4$ .

$\frac{4 (x^{4} - 256) x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4

Simplify.

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Step 2.1.1.4.1

Apply the distributive property.

$\frac{(4 x^{4} + 4 \cdot - 256) x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.2

Apply the distributive property.

$\frac{4 x^{4} x^{3} + 4 \cdot - 256 x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3

Simplify the numerator.

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Step 2.1.1.4.3.1

Simplify each term.

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Step 2.1.1.4.3.1.1

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

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Step 2.1.1.4.3.1.1.1

Move $x^{3}$ .

$\frac{4 (x^{3} x^{4}) + 4 \cdot - 256 x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3.1.1.2

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

$\frac{4 x^{3 + 4} + 4 \cdot - 256 x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3.1.1.3

Add $3$ and $4$ .

$\frac{4 x^{7} + 4 \cdot - 256 x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3.1.2

Multiply $4$ by $- 256$ .

$\frac{4 x^{7} - 1024 x^{3} - 4 x^{7}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3.2

Combine the opposite terms in $4 x^{7} - 1024 x^{3} - 4 x^{7}$ .

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Step 2.1.1.4.3.2.1

Subtract $4 x^{7}$ from $4 x^{7}$ .

$\frac{- 1024 x^{3} + 0}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.3.2.2

Add $- 1024 x^{3}$ and $0$ .

$\frac{- 1024 x^{3}}{{(x^{4} - 256)}^{2}}$

Step 2.1.1.4.4

Move the negative in front of the fraction.

$f' (x) = - \frac{1024 x^{3}}{{(x^{4} - 256)}^{2}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $- 1024$ is constant with respect to $x$ , the derivative of $- \frac{1024 x^{3}}{{(x^{4} - 256)}^{2}}$ with respect to $x$ is $- 1024 \frac{d}{d x} [\frac{x^{3}}{{(x^{4} - 256)}^{2}}]$ .

$- 1024 \frac{d}{d x} [\frac{x^{3}}{{(x^{4} - 256)}^{2}}]$

Step 2.1.2.2

$- 1024 \frac{{(x^{4} - 256)}^{2} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{4} - 256)}^{2}]}{{({(x^{4} - 256)}^{2})}^{2}}$

Step 2.1.2.3

Differentiate using the Power Rule.

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Step 2.1.2.3.1

Multiply the exponents in ${({(x^{4} - 256)}^{2})}^{2}$ .

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Step 2.1.2.3.1.1

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

$- 1024 \frac{{(x^{4} - 256)}^{2} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{4} - 256)}^{2}]}{{(x^{4} - 256)}^{2 \cdot 2}}$

Step 2.1.2.3.1.2

Multiply $2$ by $2$ .

$- 1024 \frac{{(x^{4} - 256)}^{2} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{4} - 256)}^{2}]}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.3.2

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

$- 1024 \frac{{(x^{4} - 256)}^{2} (3 x^{2}) - x^{3} \frac{d}{d x} [{(x^{4} - 256)}^{2}]}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.3.3

Move $3$ to the left of ${(x^{4} - 256)}^{2}$ .

$- 1024 \frac{3 \cdot {(x^{4} - 256)}^{2} x^{2} - x^{3} \frac{d}{d x} [{(x^{4} - 256)}^{2}]}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.4

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 $g (x) = x^{4} - 256$ .

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Step 2.1.2.4.1

To apply the Chain Rule, set $u$ as $x^{4} - 256$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - x^{3} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{4} - 256])}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.4.2

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

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - x^{3} (2 u \frac{d}{d x} [x^{4} - 256])}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.4.3

Replace all occurrences of $u$ with $x^{4} - 256$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - x^{3} (2 (x^{4} - 256) \frac{d}{d x} [x^{4} - 256])}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5

Differentiate.

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Step 2.1.2.5.1

Multiply $2$ by $- 1$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} ((x^{4} - 256) \frac{d}{d x} [x^{4} - 256])}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.2

By the Sum Rule, the derivative of $x^{4} - 256$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 256]$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} ((x^{4} - 256) (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 256]))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.3

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

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} ((x^{4} - 256) (4 x^{3} + \frac{d}{d x} [- 256]))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.4

Since $- 256$ is constant with respect to $x$ , the derivative of $- 256$ with respect to $x$ is $0$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} ((x^{4} - 256) (4 x^{3} + 0))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.5

Simplify the expression.

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Step 2.1.2.5.5.1

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

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} ((x^{4} - 256) (4 x^{3}))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.5.2

Move $4$ to the left of $x^{4} - 256$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 2 x^{3} (4 \cdot (x^{4} - 256) x^{3})}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.5.5.3

Multiply $4$ by $- 2$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{3} ((x^{4} - 256) x^{3})}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.6

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

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{3 + 3} (x^{4} - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.7

Add $3$ and $3$ .

$- 1024 \frac{3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{6} (x^{4} - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.8

Combine $- 1024$ and $\frac{3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{6} (x^{4} - 256)}{{(x^{4} - 256)}^{4}}$ .

$\frac{- 1024 (3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{6} (x^{4} - 256))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.9

Move the negative in front of the fraction.

$- \frac{1024 (3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{6} (x^{4} - 256))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10

Simplify.

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Step 2.1.2.10.1

Apply the distributive property.

$- \frac{1024 (3 {(x^{4} - 256)}^{2} x^{2} - 8 x^{6} x^{4} - 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.2

Apply the distributive property.

$- \frac{1024 (3 {(x^{4} - 256)}^{2} x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3

Simplify the numerator.

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Step 2.1.2.10.3.1

Simplify each term.

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Step 2.1.2.10.3.1.1

Rewrite ${(x^{4} - 256)}^{2}$ as $(x^{4} - 256) (x^{4} - 256)$ .

$- \frac{1024 (3 ((x^{4} - 256) (x^{4} - 256)) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.2

Expand $(x^{4} - 256) (x^{4} - 256)$ using the FOIL Method.

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Step 2.1.2.10.3.1.2.1

Apply the distributive property.

$- \frac{1024 (3 (x^{4} (x^{4} - 256) - 256 (x^{4} - 256)) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.2.2

Apply the distributive property.

$- \frac{1024 (3 (x^{4} x^{4} + x^{4} \cdot - 256 - 256 (x^{4} - 256)) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.2.3

Apply the distributive property.

$- \frac{1024 (3 (x^{4} x^{4} + x^{4} \cdot - 256 - 256 x^{4} - 256 \cdot - 256) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.3

Simplify and combine like terms.

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Step 2.1.2.10.3.1.3.1

Simplify each term.

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Step 2.1.2.10.3.1.3.1.1

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

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Step 2.1.2.10.3.1.3.1.1.1

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

$- \frac{1024 (3 (x^{4 + 4} + x^{4} \cdot - 256 - 256 x^{4} - 256 \cdot - 256) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.3.1.1.2

Add $4$ and $4$ .

$- \frac{1024 (3 (x^{8} + x^{4} \cdot - 256 - 256 x^{4} - 256 \cdot - 256) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.3.1.2

Move $- 256$ to the left of $x^{4}$ .

$- \frac{1024 (3 (x^{8} - 256 \cdot x^{4} - 256 x^{4} - 256 \cdot - 256) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.3.1.3

Multiply $- 256$ by $- 256$ .

$- \frac{1024 (3 (x^{8} - 256 x^{4} - 256 x^{4} + 65536) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.3.2

Subtract $256 x^{4}$ from $- 256 x^{4}$ .

$- \frac{1024 (3 (x^{8} - 512 x^{4} + 65536) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.4

Apply the distributive property.

$- \frac{1024 ((3 x^{8} + 3 (- 512 x^{4}) + 3 \cdot 65536) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.5

Simplify.

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Step 2.1.2.10.3.1.5.1

Multiply $- 512$ by $3$ .

$- \frac{1024 ((3 x^{8} - 1536 x^{4} + 3 \cdot 65536) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.5.2

Multiply $3$ by $65536$ .

$- \frac{1024 ((3 x^{8} - 1536 x^{4} + 196608) x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.6

Apply the distributive property.

$- \frac{1024 (3 x^{8} x^{2} - 1536 x^{4} x^{2} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7

Simplify.

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Step 2.1.2.10.3.1.7.1

Multiply $x^{8}$ by $x^{2}$ by adding the exponents.

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Step 2.1.2.10.3.1.7.1.1

Move $x^{2}$ .

$- \frac{1024 (3 (x^{2} x^{8}) - 1536 x^{4} x^{2} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7.1.2

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

$- \frac{1024 (3 x^{2 + 8} - 1536 x^{4} x^{2} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7.1.3

Add $2$ and $8$ .

$- \frac{1024 (3 x^{10} - 1536 x^{4} x^{2} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7.2

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

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Step 2.1.2.10.3.1.7.2.1

Move $x^{2}$ .

$- \frac{1024 (3 x^{10} - 1536 (x^{2} x^{4}) + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7.2.2

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

$- \frac{1024 (3 x^{10} - 1536 x^{2 + 4} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.7.2.3

Add $2$ and $4$ .

$- \frac{1024 (3 x^{10} - 1536 x^{6} + 196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.8

Apply the distributive property.

$- \frac{1024 (3 x^{10}) + 1024 (- 1536 x^{6}) + 1024 (196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.9

Simplify.

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Step 2.1.2.10.3.1.9.1

Multiply $3$ by $1024$ .

$- \frac{3072 x^{10} + 1024 (- 1536 x^{6}) + 1024 (196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.9.2

Multiply $- 1536$ by $1024$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 1024 (196608 x^{2}) + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.9.3

Multiply $196608$ by $1024$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} + 1024 (- 8 x^{6} x^{4}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.10

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 2.1.2.10.3.1.10.1

Move $x^{4}$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} + 1024 (- 8 (x^{4} x^{6})) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.10.2

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

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} + 1024 (- 8 x^{4 + 6}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.10.3

Add $4$ and $6$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} + 1024 (- 8 x^{10}) + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.11

Multiply $- 8$ by $1024$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} - 8192 x^{10} + 1024 (- 8 x^{6} \cdot - 256)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.12

Multiply $- 256$ by $- 8$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} - 8192 x^{10} + 1024 (2048 x^{6})}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.1.13

Multiply $2048$ by $1024$ .

$- \frac{3072 x^{10} - 1572864 x^{6} + 201326592 x^{2} - 8192 x^{10} + 2097152 x^{6}}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.2

Subtract $8192 x^{10}$ from $3072 x^{10}$ .

$- \frac{- 5120 x^{10} - 1572864 x^{6} + 201326592 x^{2} + 2097152 x^{6}}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.3.3

Add $- 1572864 x^{6}$ and $2097152 x^{6}$ .

$- \frac{- 5120 x^{10} + 524288 x^{6} + 201326592 x^{2}}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4

Simplify the numerator.

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Step 2.1.2.10.4.1

Factor $1024 x^{2}$ out of $- 5120 x^{10} + 524288 x^{6} + 201326592 x^{2}$ .

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Step 2.1.2.10.4.1.1

Factor $1024 x^{2}$ out of $- 5120 x^{10}$ .

$- \frac{1024 x^{2} (- 5 x^{8}) + 524288 x^{6} + 201326592 x^{2}}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.1.2

Factor $1024 x^{2}$ out of $524288 x^{6}$ .

$- \frac{1024 x^{2} (- 5 x^{8}) + 1024 x^{2} (512 x^{4}) + 201326592 x^{2}}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.1.3

Factor $1024 x^{2}$ out of $201326592 x^{2}$ .

$- \frac{1024 x^{2} (- 5 x^{8}) + 1024 x^{2} (512 x^{4}) + 1024 x^{2} (196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.1.4

Factor $1024 x^{2}$ out of $1024 x^{2} (- 5 x^{8}) + 1024 x^{2} (512 x^{4})$ .

$- \frac{1024 x^{2} (- 5 x^{8} + 512 x^{4}) + 1024 x^{2} (196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.1.5

Factor $1024 x^{2}$ out of $1024 x^{2} (- 5 x^{8} + 512 x^{4}) + 1024 x^{2} (196608)$ .

$- \frac{1024 x^{2} (- 5 x^{8} + 512 x^{4} + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.2

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

$- \frac{1024 x^{2} (- 5 {(x^{4})}^{2} + 512 x^{4} + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.3

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

$- \frac{1024 x^{2} (- 5 u^{2} + 512 u + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4

Factor by grouping.

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Step 2.1.2.10.4.4.1

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 $a \cdot c = - 5 \cdot 196608 = - 983040$ and whose sum is $b = 512$ .

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Step 2.1.2.10.4.4.1.1

Factor $512$ out of $512 u$ .

$- \frac{1024 x^{2} (- 5 u^{2} + 512 (u) + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4.1.2

Rewrite $512$ as $- 768$ plus $1280$

$- \frac{1024 x^{2} (- 5 u^{2} + (- 768 + 1280) u + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4.1.3

Apply the distributive property.

$- \frac{1024 x^{2} (- 5 u^{2} - 768 u + 1280 u + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4.2

Factor out the greatest common factor from each group.

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Step 2.1.2.10.4.4.2.1

Group the first two terms and the last two terms.

$- \frac{1024 x^{2} ((- 5 u^{2} - 768 u) + 1280 u + 196608)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4.2.2

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

$- \frac{1024 x^{2} (u (- 5 u - 768) - 256 (- 5 u - 768))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.4.3

Factor the polynomial by factoring out the greatest common factor, $- 5 u - 768$ .

$- \frac{1024 x^{2} ((- 5 u - 768) (u - 256))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.5

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

$- \frac{1024 x^{2} ((- 5 x^{4} - 768) (x^{4} - 256))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.6

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

$- \frac{1024 x^{2} ((- 5 x^{4} - 768) ({(x^{2})}^{2} - 256))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.7

Rewrite $256$ as $16^{2}$ .

$- \frac{1024 x^{2} ((- 5 x^{4} - 768) ({(x^{2})}^{2} - 16^{2}))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.8

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 $b = 16$ .

$- \frac{1024 x^{2} ((- 5 x^{4} - 768) ((x^{2} + 16) (x^{2} - 16)))}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.4.9

Factor.

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{4} - 256)}^{4}}$

Step 2.1.2.10.5

Simplify the denominator.

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Step 2.1.2.10.5.1

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

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{({(x^{2})}^{2} - 256)}^{4}}$

Step 2.1.2.10.5.2

Rewrite $256$ as $16^{2}$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{({(x^{2})}^{2} - 16^{2})}^{4}}$

Step 2.1.2.10.5.3

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 $b = 16$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x^{2} + 16) (x^{2} - 16))}^{4}}$

Step 2.1.2.10.5.4

Simplify.

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Step 2.1.2.10.5.4.1

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

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x^{2} + 16) (x^{2} - 4^{2}))}^{4}}$

Step 2.1.2.10.5.4.2

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 $b = 4$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x^{2} + 16) (x + 4) (x - 4))}^{4}}$

Step 2.1.2.10.5.5

Apply the product rule to $(x^{2} + 16) (x + 4) (x - 4)$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x^{2} + 16) (x + 4))}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.6

Expand $(x^{2} + 16) (x + 4)$ using the FOIL Method.

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Step 2.1.2.10.5.6.1

Apply the distributive property.

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2} (x + 4) + 16 (x + 4))}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.6.2

Apply the distributive property.

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2} x + x^{2} \cdot 4 + 16 (x + 4))}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.6.3

Apply the distributive property.

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2} x + x^{2} \cdot 4 + 16 x + 16 \cdot 4)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.7

Simplify each term.

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Step 2.1.2.10.5.7.1

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

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Step 2.1.2.10.5.7.1.1

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

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Step 2.1.2.10.5.7.1.1.1

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

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2} x^{1} + x^{2} \cdot 4 + 16 x + 16 \cdot 4)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.7.1.1.2

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

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2 + 1} + x^{2} \cdot 4 + 16 x + 16 \cdot 4)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.7.1.2

Add $2$ and $1$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{3} + x^{2} \cdot 4 + 16 x + 16 \cdot 4)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.7.2

Move $4$ to the left of $x^{2}$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{3} + 4 \cdot x^{2} + 16 x + 16 \cdot 4)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.7.3

Multiply $16$ by $4$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{3} + 4 x^{2} + 16 x + 64)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.8

Factor out the greatest common factor from each group.

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Step 2.1.2.10.5.8.1

Group the first two terms and the last two terms.

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x^{3} + 4 x^{2}) + 16 x + 64)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.8.2

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

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x^{2} (x + 4) + 16 (x + 4))}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.9

Factor the polynomial by factoring out the greatest common factor, $x + 4$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{((x + 4) (x^{2} + 16))}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.5.10

Apply the product rule to $(x + 4) (x^{2} + 16)$ .

$- \frac{1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)}{{(x + 4)}^{4} {(x^{2} + 16)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.6

Cancel the common factor of $x^{2} + 16$ and ${(x^{2} + 16)}^{4}$ .

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Step 2.1.2.10.6.1

Factor $x^{2} + 16$ out of $1024 x^{2} (- 5 x^{4} - 768) (x^{2} + 16) (x + 4) (x - 4)$ .

$- \frac{(x^{2} + 16) ((1024 x^{2} (- 5 x^{4} - 768) (x + 4)) (x - 4))}{{(x + 4)}^{4} {(x^{2} + 16)}^{4} {(x - 4)}^{4}}$

Step 2.1.2.10.6.2

Cancel the common factors.

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Step 2.1.2.10.6.2.1

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

$- \frac{(x^{2} + 16) ((1024 x^{2} (- 5 x^{4} - 768) (x + 4)) (x - 4))}{(x^{2} + 16) ({(x + 4)}^{4} {(x^{2} + 16)}^{3} {(x - 4)}^{4})}$

Step 2.1.2.10.6.2.2

Cancel the common factor.

$- \frac{(x^{2} + 16) ((1024 x^{2} (- 5 x^{4} - 768) (x + 4)) (x - 4))}{(x^{2} + 16) ({(x + 4)}^{4} {(x^{2} + 16)}^{3} {(x - 4)}^{4})}$

Step 2.1.2.10.6.2.3

Rewrite the expression.

$- \frac{(1024 x^{2} (- 5 x^{4} - 768) (x + 4)) (x - 4)}{{(x + 4)}^{4} {(x^{2} + 16)}^{3} {(x - 4)}^{4}}$

Step 2.1.2.10.7

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

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Step 2.1.2.10.7.1

Factor $x + 4$ out of $(1024 x^{2} (- 5 x^{4} - 768) (x + 4)) (x - 4)$ .

$- \frac{(x + 4) ((1024 x^{2} (- 5 x^{4} - 768)) (x - 4))}{{(x + 4)}^{4} {(x^{2} + 16)}^{3} {(x - 4)}^{4}}$

Step 2.1.2.10.7.2

Cancel the common factors.

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Step 2.1.2.10.7.2.1

Factor $x + 4$ out of ${(x + 4)}^{4} {(x^{2} + 16)}^{3} {(x - 4)}^{4}$ .

$- \frac{(x + 4) ((1024 x^{2} (- 5 x^{4} - 768)) (x - 4))}{(x + 4) ({(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{4})}$

Step 2.1.2.10.7.2.2

Cancel the common factor.

$- \frac{(x + 4) ((1024 x^{2} (- 5 x^{4} - 768)) (x - 4))}{(x + 4) ({(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{4})}$

Step 2.1.2.10.7.2.3

Rewrite the expression.

$- \frac{(1024 x^{2} (- 5 x^{4} - 768)) (x - 4)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{4}}$

Step 2.1.2.10.8

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

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Step 2.1.2.10.8.1

Factor $x - 4$ out of $(1024 x^{2} (- 5 x^{4} - 768)) (x - 4)$ .

$- \frac{(x - 4) (1024 x^{2} (- 5 x^{4} - 768))}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{4}}$

Step 2.1.2.10.8.2

Cancel the common factors.

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Step 2.1.2.10.8.2.1

Factor $x - 4$ out of ${(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{4}$ .

$- \frac{(x - 4) (1024 x^{2} (- 5 x^{4} - 768))}{(x - 4) ({(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3})}$

Step 2.1.2.10.8.2.2

Cancel the common factor.

$- \frac{(x - 4) (1024 x^{2} (- 5 x^{4} - 768))}{(x - 4) ({(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3})}$

Step 2.1.2.10.8.2.3

Rewrite the expression.

$- \frac{1024 x^{2} (- 5 x^{4} - 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.9

Factor $- 1$ out of $- 5 x^{4}$ .

$- \frac{1024 x^{2} (- (5 x^{4}) - 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.10

Rewrite $- 768$ as $- 1 (768)$ .

$- \frac{1024 x^{2} (- (5 x^{4}) - 1 (768))}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.11

Factor $- 1$ out of $- (5 x^{4}) - 1 (768)$ .

$- \frac{1024 x^{2} (- (5 x^{4} + 768))}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.12

Rewrite $- (5 x^{4} + 768)$ as $- 1 (5 x^{4} + 768)$ .

$- \frac{1024 x^{2} (- 1 (5 x^{4} + 768))}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.13

Move the negative in front of the fraction.

$- - \frac{(1024 x^{2}) (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.14

Multiply $- 1$ by $- 1$ .

$1 \frac{(1024 x^{2}) (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.15

Multiply $\frac{(1024 x^{2}) (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$ by $1$ .

$\frac{(1024 x^{2}) (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.2.10.16

Reorder factors in $\frac{(1024 x^{2}) (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$ .

$f'' (x) = \frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$ .

$\frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$1024 x^{2} (5 x^{4} + 768) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$x^{2} = 0$

$5 x^{4} + 768 = 0$

Step 2.2.3.2

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

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Step 2.2.3.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.2.3.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.2.3.2.2.1

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

$x = \pm \sqrt{0}$

Step 2.2.3.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.3.2.2.2.1

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

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

Step 2.2.3.2.2.2.2

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

$x = \pm 0$

Step 2.2.3.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.3.3

Set $5 x^{4} + 768$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $5 x^{4} + 768$ equal to $0$ .

$5 x^{4} + 768 = 0$

Step 2.2.3.3.2

Solve $5 x^{4} + 768 = 0$ for $x$ .

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Step 2.2.3.3.2.1

Subtract $768$ from both sides of the equation.

$5 x^{4} = - 768$

Step 2.2.3.3.2.2

Divide each term in $5 x^{4} = - 768$ by $5$ and simplify.

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Step 2.2.3.3.2.2.1

Divide each term in $5 x^{4} = - 768$ by $5$ .

$\frac{5 x^{4}}{5} = \frac{- 768}{5}$

Step 2.2.3.3.2.2.2

Simplify the left side.

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Step 2.2.3.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{4}}{5} = \frac{- 768}{5}$

Step 2.2.3.3.2.2.2.1.2

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

$x^{4} = \frac{- 768}{5}$

Step 2.2.3.3.2.2.3

Simplify the right side.

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Step 2.2.3.3.2.2.3.1

Move the negative in front of the fraction.

$x^{4} = - \frac{768}{5}$

Step 2.2.3.3.2.3

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

$x = \pm \sqrt[4]{- \frac{768}{5}}$

Step 2.2.3.3.2.4

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

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Step 2.2.3.3.2.4.1

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

$x = \sqrt[4]{- \frac{768}{5}}$

Step 2.2.3.3.2.4.2

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

$x = - \sqrt[4]{- \frac{768}{5}}$

Step 2.2.3.3.2.4.3

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

$x = \sqrt[4]{- \frac{768}{5}}, - \sqrt[4]{- \frac{768}{5}}$

Step 2.2.3.4

The final solution is all the values that make $1024 x^{2} (5 x^{4} + 768) = 0$ true.

$x = 0, \sqrt[4]{- \frac{768}{5}}, - \sqrt[4]{- \frac{768}{5}}$

Step 2.2.4

Exclude the solutions that do not make $\frac{1024 x^{2} (5 x^{4} + 768)}{{(x + 4)}^{3} {(x^{2} + 16)}^{3} {(x - 4)}^{3}} = 0$ true.

$x = 0$

Step 3

Find the domain of $f (x) = \frac{x^{4}}{x^{4} - 256}$ .

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Step 3.1

Set the denominator in $\frac{x^{4}}{x^{4} - 256}$ equal to $0$ to find where the expression is undefined.

$x^{4} - 256 = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Add $256$ to both sides of the equation.

$x^{4} = 256$

Step 3.2.2

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

$x = \pm \sqrt[4]{256}$

Step 3.2.3

Simplify $\pm \sqrt[4]{256}$ .

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Step 3.2.3.1

Rewrite $256$ as $4^{4}$ .

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

Step 3.2.3.2

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

$x = \pm 4$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = 4$

Step 3.2.4.2

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

$x = - 4$

Step 3.2.4.3

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

$x = 4, - 4$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 4, - 4}$

Interval Notation:

$(- \infty, - 4) \cup (- 4, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 4, - 4}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 4) \cup (- 4, 0) \cup (0, 4) \cup (4, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 6$ in the expression.

$f'' (- 6) = \frac{1024 {(- 6)}^{2} (5 {(- 6)}^{4} + 768)}{{((- 6) + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {((- 6) - 4)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Raise $- 6$ to the power of $4$ .

$f'' (- 6) = \frac{1024 {(- 6)}^{2} (5 \cdot 1296 + 768)}{{(- 6 + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.1.2

Multiply $5$ by $1296$ .

$f'' (- 6) = \frac{1024 {(- 6)}^{2} (6480 + 768)}{{(- 6 + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.1.3

Add $6480$ and $768$ .

$f'' (- 6) = \frac{1024 {(- 6)}^{2} \cdot 7248}{{(- 6 + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.1.4

Multiply $7248$ by $1024$ .

$f'' (- 6) = \frac{7421952 {(- 6)}^{2}}{{(- 6 + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.1.5

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{{(- 6 + 4)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add $- 6$ and $4$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{{(- 2)}^{3} {({(- 6)}^{2} + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.2.2

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{{(- 2)}^{3} {(36 + 16)}^{3} {(- 6 - 4)}^{3}}$

Step 5.2.2.3

Add $36$ and $16$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{{(- 2)}^{3} \cdot (52^{3} {(- 6 - 4)}^{3})}$

Step 5.2.2.4

Subtract $4$ from $- 6$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{{(- 2)}^{3} \cdot (52^{3} {(- 10)}^{3})}$

Step 5.2.2.5

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

$f'' (- 6) = \frac{7421952 \cdot 36}{- 8 \cdot (52^{3} {(- 10)}^{3})}$

Step 5.2.2.6

Raise $52$ to the power of $3$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{- 8 \cdot (140608 {(- 10)}^{3})}$

Step 5.2.2.7

Raise $- 10$ to the power of $3$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{- 8 \cdot 140608 \cdot - 1000}$

Step 5.2.2.8

Combine exponents.

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Step 5.2.2.8.1

Multiply $- 8$ by $140608$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{- 1124864 \cdot - 1000}$

Step 5.2.2.8.2

Multiply $- 1124864$ by $- 1000$ .

$f'' (- 6) = \frac{7421952 \cdot 36}{1124864000}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Multiply $7421952$ by $36$ .

$f'' (- 6) = \frac{267190272}{1124864000}$

Step 5.2.3.2

Cancel the common factor of $267190272$ and $1124864000$ .

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Step 5.2.3.2.1

Factor $4096$ out of $267190272$ .

$f'' (- 6) = \frac{4096 (65232)}{1124864000}$

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

Factor $4096$ out of $1124864000$ .

$f'' (- 6) = \frac{4096 \cdot 65232}{4096 \cdot 274625}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f'' (- 6) = \frac{4096 \cdot 65232}{4096 \cdot 274625}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f'' (- 6) = \frac{65232}{274625}$

Step 5.2.4

The final answer is $\frac{65232}{274625}$ .

$\frac{65232}{274625}$

Step 5.3

The graph is concave up on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 4, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 2$ in the expression.

$f'' (- 2) = \frac{1024 {(- 2)}^{2} (5 {(- 2)}^{4} + 768)}{{((- 2) + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {((- 2) - 4)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

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

$f'' (- 2) = \frac{1024 {(- 2)}^{2} (5 \cdot 16 + 768)}{{(- 2 + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.1.2

Multiply $5$ by $16$ .

$f'' (- 2) = \frac{1024 {(- 2)}^{2} (80 + 768)}{{(- 2 + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.1.3

Add $80$ and $768$ .

$f'' (- 2) = \frac{1024 {(- 2)}^{2} \cdot 848}{{(- 2 + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.1.4

Multiply $848$ by $1024$ .

$f'' (- 2) = \frac{868352 {(- 2)}^{2}}{{(- 2 + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.1.5

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

$f'' (- 2) = \frac{868352 \cdot 4}{{(- 2 + 4)}^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 2$ and $4$ .

$f'' (- 2) = \frac{868352 \cdot 4}{2^{3} {({(- 2)}^{2} + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.2.2

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

$f'' (- 2) = \frac{868352 \cdot 4}{2^{3} {(4 + 16)}^{3} {(- 2 - 4)}^{3}}$

Step 6.2.2.3

Add $4$ and $16$ .

$f'' (- 2) = \frac{868352 \cdot 4}{2^{3} \cdot (20^{3} {(- 2 - 4)}^{3})}$

Step 6.2.2.4

Subtract $4$ from $- 2$ .

$f'' (- 2) = \frac{868352 \cdot 4}{2^{3} \cdot (20^{3} {(- 6)}^{3})}$

Step 6.2.2.5

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

$f'' (- 2) = \frac{868352 \cdot 4}{8 \cdot (20^{3} {(- 6)}^{3})}$

Step 6.2.2.6

Raise $20$ to the power of $3$ .

$f'' (- 2) = \frac{868352 \cdot 4}{8 \cdot (8000 {(- 6)}^{3})}$

Step 6.2.2.7

Raise $- 6$ to the power of $3$ .

$f'' (- 2) = \frac{868352 \cdot 4}{8 \cdot 8000 \cdot - 216}$

Step 6.2.2.8

Combine exponents.

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Step 6.2.2.8.1

Multiply $8$ by $8000$ .

$f'' (- 2) = \frac{868352 \cdot 4}{64000 \cdot - 216}$

Step 6.2.2.8.2

Multiply $64000$ by $- 216$ .

$f'' (- 2) = \frac{868352 \cdot 4}{- 13824000}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $868352$ by $4$ .

$f'' (- 2) = \frac{3473408}{- 13824000}$

Step 6.2.3.2

Cancel the common factor of $3473408$ and $- 13824000$ .

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Step 6.2.3.2.1

Factor $4096$ out of $3473408$ .

$f'' (- 2) = \frac{4096 (848)}{- 13824000}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $4096$ out of $- 13824000$ .

$f'' (- 2) = \frac{4096 \cdot 848}{4096 \cdot - 3375}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (- 2) = \frac{4096 \cdot 848}{4096 \cdot - 3375}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (- 2) = \frac{848}{- 3375}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f'' (- 2) = - \frac{848}{3375}$

Step 6.2.4

The final answer is $- \frac{848}{3375}$ .

$- \frac{848}{3375}$

Step 6.3

The graph is concave down on the interval $(- 4, 0)$ because $f'' (- 2)$ is negative.

Concave down on $(- 4, 0)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(0, 4)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = \frac{1024 {(2)}^{2} (5 {(2)}^{4} + 768)}{{((2) + 4)}^{3} {({(2)}^{2} + 16)}^{3} {((2) - 4)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Rewrite $1024$ as $2^{10}$ .

$f'' (2) = \frac{2^{10} \cdot (2^{2} (5 \cdot 2^{4} + 768))}{{(2 + 4)}^{3} {(2^{2} + 16)}^{3} {(2 - 4)}^{3}}$

Step 7.2.1.2

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

$f'' (2) = \frac{2^{10 + 2} (5 \cdot 2^{4} + 768)}{{(2 + 4)}^{3} {(2^{2} + 16)}^{3} {(2 - 4)}^{3}}$

Step 7.2.1.3

Add $10$ and $2$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{{(2 + 4)}^{3} {(2^{2} + 16)}^{3} {(2 - 4)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $2$ and $4$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{6^{3} {(2^{2} + 16)}^{3} {(2 - 4)}^{3}}$

Step 7.2.2.2

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

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{6^{3} {(4 + 16)}^{3} {(2 - 4)}^{3}}$

Step 7.2.2.3

Add $4$ and $16$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{6^{3} \cdot (20^{3} {(2 - 4)}^{3})}$

Step 7.2.2.4

Subtract $4$ from $2$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{6^{3} \cdot (20^{3} {(- 2)}^{3})}$

Step 7.2.2.5

Raise $6$ to the power of $3$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{216 \cdot (20^{3} {(- 2)}^{3})}$

Step 7.2.2.6

Raise $20$ to the power of $3$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{216 \cdot (8000 {(- 2)}^{3})}$

Step 7.2.2.7

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

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{216 \cdot 8000 \cdot - 8}$

Step 7.2.2.8

Combine exponents.

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Step 7.2.2.8.1

Multiply $216$ by $8000$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{1728000 \cdot - 8}$

Step 7.2.2.8.2

Multiply $1728000$ by $- 8$ .

$f'' (2) = \frac{2^{12} (5 \cdot 2^{4} + 768)}{- 13824000}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

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

$f'' (2) = \frac{2^{12} (5 \cdot 16 + 768)}{- 13824000}$

Step 7.2.3.2

Multiply $5$ by $16$ .

$f'' (2) = \frac{2^{12} (80 + 768)}{- 13824000}$

Step 7.2.3.3

Add $80$ and $768$ .

$f'' (2) = \frac{2^{12} \cdot 848}{- 13824000}$

Step 7.2.3.4

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

$f'' (2) = \frac{4096 \cdot 848}{- 13824000}$

Step 7.2.4

Reduce the expression by cancelling the common factors.

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Step 7.2.4.1

Multiply $4096$ by $848$ .

$f'' (2) = \frac{3473408}{- 13824000}$

Step 7.2.4.2

Cancel the common factor of $3473408$ and $- 13824000$ .

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Step 7.2.4.2.1

Factor $4096$ out of $3473408$ .

$f'' (2) = \frac{4096 (848)}{- 13824000}$

Step 7.2.4.2.2

Cancel the common factors.

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Step 7.2.4.2.2.1

Factor $4096$ out of $- 13824000$ .

$f'' (2) = \frac{4096 \cdot 848}{4096 \cdot - 3375}$

Step 7.2.4.2.2.2

Cancel the common factor.

$f'' (2) = \frac{4096 \cdot 848}{4096 \cdot - 3375}$

Step 7.2.4.2.2.3

Rewrite the expression.

$f'' (2) = \frac{848}{- 3375}$

Step 7.2.4.3

Move the negative in front of the fraction.

$f'' (2) = - \frac{848}{3375}$

Step 7.2.5

The final answer is $- \frac{848}{3375}$ .

$- \frac{848}{3375}$

Step 7.3

The graph is concave down on the interval $(0, 4)$ because $f'' (2)$ is negative.

Concave down on $(0, 4)$ since $f'' (x)$ is negative

Step 8

Substitute any number from the interval $(4, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $6$ in the expression.

$f'' (6) = \frac{1024 {(6)}^{2} (5 {(6)}^{4} + 768)}{{((6) + 4)}^{3} {({(6)}^{2} + 16)}^{3} {((6) - 4)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Raise $6$ to the power of $4$ .

$f'' (6) = \frac{1024 \cdot (6^{2} (5 \cdot 1296 + 768))}{{(6 + 4)}^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.1.2

Multiply $5$ by $1296$ .

$f'' (6) = \frac{1024 \cdot (6^{2} (6480 + 768))}{{(6 + 4)}^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.1.3

Add $6480$ and $768$ .

$f'' (6) = \frac{1024 \cdot 6^{2} \cdot 7248}{{(6 + 4)}^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.1.4

Multiply $7248$ by $1024$ .

$f'' (6) = \frac{7421952 \cdot 6^{2}}{{(6 + 4)}^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.1.5

Raise $6$ to the power of $2$ .

$f'' (6) = \frac{7421952 \cdot 36}{{(6 + 4)}^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $6$ and $4$ .

$f'' (6) = \frac{7421952 \cdot 36}{10^{3} {(6^{2} + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.2.2

Raise $6$ to the power of $2$ .

$f'' (6) = \frac{7421952 \cdot 36}{10^{3} {(36 + 16)}^{3} {(6 - 4)}^{3}}$

Step 8.2.2.3

Add $36$ and $16$ .

$f'' (6) = \frac{7421952 \cdot 36}{10^{3} \cdot (52^{3} {(6 - 4)}^{3})}$

Step 8.2.2.4

Subtract $4$ from $6$ .

$f'' (6) = \frac{7421952 \cdot 36}{10^{3} \cdot 52^{3} \cdot 2^{3}}$

Step 8.2.2.5

Raise $10$ to the power of $3$ .

$f'' (6) = \frac{7421952 \cdot 36}{1000 \cdot 52^{3} \cdot 2^{3}}$

Step 8.2.2.6

Raise $52$ to the power of $3$ .

$f'' (6) = \frac{7421952 \cdot 36}{1000 \cdot 140608 \cdot 2^{3}}$

Step 8.2.2.7

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

$f'' (6) = \frac{7421952 \cdot 36}{1000 \cdot 140608 \cdot 8}$

Step 8.2.2.8

Combine exponents.

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Step 8.2.2.8.1

Multiply $1000$ by $140608$ .

$f'' (6) = \frac{7421952 \cdot 36}{140608000 \cdot 8}$

Step 8.2.2.8.2

Multiply $140608000$ by $8$ .

$f'' (6) = \frac{7421952 \cdot 36}{1124864000}$

Step 8.2.3

Reduce the expression by cancelling the common factors.

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Step 8.2.3.1

Multiply $7421952$ by $36$ .

$f'' (6) = \frac{267190272}{1124864000}$

Step 8.2.3.2

Cancel the common factor of $267190272$ and $1124864000$ .

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Step 8.2.3.2.1

Factor $4096$ out of $267190272$ .

$f'' (6) = \frac{4096 (65232)}{1124864000}$

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $4096$ out of $1124864000$ .

$f'' (6) = \frac{4096 \cdot 65232}{4096 \cdot 274625}$

Step 8.2.3.2.2.2

Cancel the common factor.

$f'' (6) = \frac{4096 \cdot 65232}{4096 \cdot 274625}$

Step 8.2.3.2.2.3

Rewrite the expression.

$f'' (6) = \frac{65232}{274625}$

Step 8.2.4

The final answer is $\frac{65232}{274625}$ .

$\frac{65232}{274625}$

Step 8.3

The graph is concave up on the interval $(4, \infty)$ because $f'' (6)$ is positive.

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 9

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Concave down on $(- 4, 0)$ since $f'' (x)$ is negative

Concave down on $(0, 4)$ since $f'' (x)$ is negative

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 10