Calculus Examples

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

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = x^{2} + 1$ and $g (x) = x^{2} - 4$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.4.2

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

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

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

Step 1.1.1.2.7

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 1.1.1.2.8

Simplify the expression.

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Step 1.1.1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.3.3

Apply the distributive property.

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

Step 1.1.1.3.4

Apply the distributive property.

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

Step 1.1.1.3.5

Simplify the numerator.

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Step 1.1.1.3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot - 4 x - 2 x^{2} x - 2 \cdot 1 x$ .

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Step 1.1.1.3.5.1.1

Subtract $2 x^{2} x$ from $2 x^{2} x$ .

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

Step 1.1.1.3.5.1.2

Add $2 \cdot - 4 x$ and $0$ .

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

Step 1.1.1.3.5.2

Simplify each term.

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Step 1.1.1.3.5.2.1

Multiply $2$ by $- 4$ .

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

Step 1.1.1.3.5.2.2

Multiply $- 2$ by $1$ .

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

Step 1.1.1.3.5.3

Subtract $2 x$ from $- 8 x$ .

$f' (x) = \frac{- 10 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.1.3.6

Move the negative in front of the fraction.

$f' (x) = - \frac{10 x}{{(x^{2} - 4)}^{2}}$

Step 1.1.1.3.7

Simplify the denominator.

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Step 1.1.1.3.7.1

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

$f' (x) = - \frac{10 x}{{(x^{2} - 2^{2})}^{2}}$

Step 1.1.1.3.7.2

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 $b = 2$ .

$f' (x) = - \frac{10 x}{{((x + 2) (x - 2))}^{2}}$

Step 1.1.1.3.7.3

Apply the product rule to $(x + 2) (x - 2)$ .

$f' (x) = - \frac{10 x}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10 x}{{(x + 2)}^{2} {(x - 2)}^{2}}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{x}{{(x + 2)}^{2} {(x - 2)}^{2}}]$ .

$f'' (x) = - 10 \frac{d}{d x} \frac{x}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.1.2.2

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} ({(x + 2)}^{2} {(x - 2)}^{2})}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.3

Differentiate using the Power Rule.

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Step 1.1.2.3.1

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} \cdot 1 - x \frac{d}{d x} ({(x + 2)}^{2} {(x - 2)}^{2})}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.3.2

Multiply ${(x + 2)}^{2}$ by $1$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x \frac{d}{d x} ({(x + 2)}^{2} {(x - 2)}^{2})}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.4

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x ({(x + 2)}^{2} \frac{d}{d x} ({(x - 2)}^{2}) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.5

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 $g (x) = x - 2$ .

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Step 1.1.2.5.1

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x ({(x + 2)}^{2} (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (x - 2)) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.5.2

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x ({(x + 2)}^{2} (2 u_{1} \frac{d}{d x} (x - 2)) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.5.3

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x ({(x + 2)}^{2} (2 (x - 2) \frac{d}{d x} (x - 2)) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6

Differentiate.

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Step 1.1.2.6.1

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 \cdot ({(x + 2)}^{2} (x - 2)) \frac{d}{d x} (x - 2) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6.2

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) (\frac{d}{d x} (x) + \frac{d}{d x} (- 2)) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6.3

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) (1 + \frac{d}{d x} (- 2)) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) (1 + 0) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6.5

Simplify the expression.

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Step 1.1.2.6.5.1

Add $1$ and $0$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) \cdot 1 + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.6.5.2

Multiply $2$ by $1$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} \frac{d}{d x} ({(x + 2)}^{2}))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.7

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 $g (x) = x + 2$ .

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Step 1.1.2.7.1

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (x + 2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.7.2

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (2 u_{2} \frac{d}{d x} (x + 2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.7.3

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (2 (x + 2) \frac{d}{d x} (x + 2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8

Differentiate.

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Step 1.1.2.8.1

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 \cdot ({(x - 2)}^{2} (x + 2)) \frac{d}{d x} (x + 2))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.2

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (\frac{d}{d x} (x) + \frac{d}{d x} (2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.3

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

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (1 + \frac{d}{d x} (2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (1 + 0))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.5

Combine fractions.

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Step 1.1.2.8.5.1

Add $1$ and $0$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) \cdot 1)}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.5.2

Multiply $2$ by $1$ .

$f'' (x) = - 10 \frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.5.3

Combine $- 10$ and $\frac{{(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$ .

$f'' (x) = \frac{- 10 ({(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.8.5.4

Move the negative in front of the fraction.

$f'' (x) = - \frac{(10) ({(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 1.1.2.9

Simplify.

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Step 1.1.2.9.1

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

$f'' (x) = - \frac{10 ({(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.2

Apply the distributive property.

$f'' (x) = - \frac{10 ({(x + 2)}^{2} {(x - 2)}^{2} - x (2 {(x + 2)}^{2} (x - 2)) - x (2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.3

Apply the distributive property.

$f'' (x) = - \frac{10 ({(x + 2)}^{2} {(x - 2)}^{2}) + 10 (- x (2 {(x + 2)}^{2} (x - 2))) + 10 (- x (2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4

Simplify the numerator.

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Step 1.1.2.9.4.1

Factor $10 (x + 2) (x - 2)$ out of $10 {(x + 2)}^{2} {(x - 2)}^{2} + 10 (- x (2 {(x + 2)}^{2} (x - 2))) + 10 (- x (2 {(x - 2)}^{2} (x + 2)))$ .

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Step 1.1.2.9.4.1.1

Factor $10 (x + 2) (x - 2)$ out of $10 {(x + 2)}^{2} {(x - 2)}^{2}$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2)) + 10 (- x (2 {(x + 2)}^{2} (x - 2))) + 10 (- x (2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.1.2

Factor $10 (x + 2) (x - 2)$ out of $10 (- x (2 {(x + 2)}^{2} (x - 2)))$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2)) + 10 (x + 2) (x - 2) (- x (2 (x + 2))) + 10 (- x (2 {(x - 2)}^{2} (x + 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.1.3

Factor $10 (x + 2) (x - 2)$ out of $10 (- x (2 {(x - 2)}^{2} (x + 2)))$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2)) + 10 (x + 2) (x - 2) (- x (2 (x + 2))) + 10 (x + 2) (x - 2) (- x (2 (x - 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.1.4

Factor $10 (x + 2) (x - 2)$ out of $10 (x + 2) (x - 2) ((x + 2) (x - 2)) + 10 (x + 2) (x - 2) (- x (2 (x + 2)))$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2) - x (2 (x + 2))) + 10 (x + 2) (x - 2) (- x (2 (x - 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.1.5

Factor $10 (x + 2) (x - 2)$ out of $10 (x + 2) (x - 2) ((x + 2) (x - 2) - x (2 (x + 2))) + 10 (x + 2) (x - 2) (- x (2 (x - 2)))$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2) - x (2 (x + 2)) - x (2 (x - 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.2

Combine exponents.

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Step 1.1.2.9.4.2.1

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2) - 2 x (x + 2) - x (2 (x - 2)))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) ((x + 2) (x - 2) - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3

Simplify each term.

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Step 1.1.2.9.4.3.1

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 1.1.2.9.4.3.1.1

Apply the distributive property.

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x (x - 2) + 2 (x - 2) - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.1.2

Apply the distributive property.

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x \cdot x + x \cdot - 2 + 2 (x - 2) - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.1.3

Apply the distributive property.

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.2

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 1.1.2.9.4.3.2.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x \cdot x - 2 x + 2 x + 2 \cdot - 2 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.2.2

Add $- 2 x$ and $2 x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x \cdot x + 0 + 2 \cdot - 2 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.2.3

Add $x \cdot x$ and $0$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x \cdot x + 2 \cdot - 2 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.3

Simplify each term.

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Step 1.1.2.9.4.3.3.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} + 2 \cdot - 2 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.3.2

Multiply $2$ by $- 2$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x (x + 2) - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.4

Apply the distributive property.

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x \cdot x - 2 x \cdot 2 - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.2.9.4.3.5.1

Move $x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 (x \cdot x) - 2 x \cdot 2 - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 2 x \cdot 2 - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.6

Multiply $2$ by $- 2$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 4 x - 2 x (x - 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.7

Apply the distributive property.

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 4 x - 2 x \cdot x - 2 x \cdot - 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.8

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.2.9.4.3.8.1

Move $x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 4 x - 2 (x \cdot x) - 2 x \cdot - 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.8.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 4 x - 2 x^{2} - 2 x \cdot - 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.3.9

Multiply $- 2$ by $- 2$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 4 x - 2 x^{2} + 4 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.4

Combine the opposite terms in $x^{2} - 4 - 2 x^{2} - 4 x - 2 x^{2} + 4 x$ .

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Step 1.1.2.9.4.4.1

Add $- 4 x$ and $4 x$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 2 x^{2} + 0)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.4.2

Add $x^{2} - 4 - 2 x^{2} - 2 x^{2}$ and $0$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (x^{2} - 4 - 2 x^{2} - 2 x^{2})}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.5

Subtract $2 x^{2}$ from $x^{2}$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (- x^{2} - 4 - 2 x^{2})}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.4.6

Subtract $2 x^{2}$ from $- x^{2}$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (- 3 x^{2} - 4)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.5

Combine terms.

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Step 1.1.2.9.5.1

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

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Step 1.1.2.9.5.1.1

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

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

Step 1.1.2.9.5.1.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (- 3 x^{2} - 4)}{{(x + 2)}^{4} {({(x - 2)}^{2})}^{2}}$

Step 1.1.2.9.5.2

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

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Step 1.1.2.9.5.2.1

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

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

Step 1.1.2.9.5.2.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{10 (x + 2) (x - 2) (- 3 x^{2} - 4)}{{(x + 2)}^{4} {(x - 2)}^{4}}$

Step 1.1.2.9.5.3

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

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Step 1.1.2.9.5.3.1

Factor $x + 2$ out of $10 (x + 2) (x - 2) (- 3 x^{2} - 4)$ .

$f'' (x) = - \frac{(x + 2) ((10 (x - 2)) (- 3 x^{2} - 4))}{{(x + 2)}^{4} {(x - 2)}^{4}}$

Step 1.1.2.9.5.3.2

Cancel the common factors.

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Step 1.1.2.9.5.3.2.1

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

$f'' (x) = - \frac{(x + 2) ((10 (x - 2)) (- 3 x^{2} - 4))}{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{4})}$

Step 1.1.2.9.5.3.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x + 2) ((10 (x - 2)) (- 3 x^{2} - 4))}{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{4})}$

Step 1.1.2.9.5.3.2.3

Rewrite the expression.

$f'' (x) = - \frac{(10 (x - 2)) (- 3 x^{2} - 4)}{{(x + 2)}^{3} {(x - 2)}^{4}}$

Step 1.1.2.9.5.4

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

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Step 1.1.2.9.5.4.1

Factor $x - 2$ out of $10 (x - 2) (- 3 x^{2} - 4)$ .

$f'' (x) = - \frac{(x - 2) (10 (- 3 x^{2} - 4))}{{(x + 2)}^{3} {(x - 2)}^{4}}$

Step 1.1.2.9.5.4.2

Cancel the common factors.

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Step 1.1.2.9.5.4.2.1

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

$f'' (x) = - \frac{(x - 2) (10 (- 3 x^{2} - 4))}{(x - 2) ({(x + 2)}^{3} {(x - 2)}^{3})}$

Step 1.1.2.9.5.4.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x - 2) (10 (- 3 x^{2} - 4))}{(x - 2) ({(x + 2)}^{3} {(x - 2)}^{3})}$

Step 1.1.2.9.5.4.2.3

Rewrite the expression.

$f'' (x) = - \frac{10 (- 3 x^{2} - 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.6

Factor $- 1$ out of $- 3 x^{2}$ .

$f'' (x) = - \frac{10 (- (3 x^{2}) - 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.7

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

$f'' (x) = - \frac{10 (- (3 x^{2}) - 1 \cdot 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.8

Factor $- 1$ out of $- (3 x^{2}) - 1 (4)$ .

$f'' (x) = - \frac{10 (- (3 x^{2} + 4))}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.9

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

$f'' (x) = - \frac{10 (- 1 (3 x^{2} + 4))}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.10

Move the negative in front of the fraction.

$f'' (x) = \frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.2.9.11

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}})$

Step 1.1.2.9.12

Multiply $\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$ by $1$ .

$f'' (x) = \frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$ .

$\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{10 (3 x^{2} + 4)}{{(x + 2)}^{3} {(x - 2)}^{3}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$10 (3 x^{2} + 4) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $10 (3 x^{2} + 4) = 0$ by $10$ and simplify.

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Step 1.2.3.1.1

Divide each term in $10 (3 x^{2} + 4) = 0$ by $10$ .

$\frac{10 (3 x^{2} + 4)}{10} = \frac{0}{10}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $10$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{10 (3 x^{2} + 4)}{10} = \frac{0}{10}$

Step 1.2.3.1.2.1.2

Divide $3 x^{2} + 4$ by $1$ .

$3 x^{2} + 4 = \frac{0}{10}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $10$ .

$3 x^{2} + 4 = 0$

Step 1.2.3.2

Subtract $4$ from both sides of the equation.

$3 x^{2} = - 4$

Step 1.2.3.3

Divide each term in $3 x^{2} = - 4$ by $3$ and simplify.

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Step 1.2.3.3.1

Divide each term in $3 x^{2} = - 4$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 4}{3}$

Step 1.2.3.3.2

Simplify the left side.

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Step 1.2.3.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 4}{3}$

Step 1.2.3.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{3}$

Step 1.2.3.3.3

Simplify the right side.

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Step 1.2.3.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{4}{3}$

Step 1.2.3.4

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

$x = \pm \sqrt{- \frac{4}{3}}$

Step 1.2.3.5

Simplify $\pm \sqrt{- \frac{4}{3}}$ .

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Step 1.2.3.5.1

Rewrite $- \frac{4}{3}$ as $i^{2} \frac{2^{2}}{3}$ .

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Step 1.2.3.5.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{4}{3}}$

Step 1.2.3.5.1.2

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

$x = \pm \sqrt{i^{2} \frac{2^{2}}{3}}$

Step 1.2.3.5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2^{2}}{3}}$

Step 1.2.3.5.3

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

$x = \pm i \sqrt{\frac{4}{3}}$

Step 1.2.3.5.4

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{4}}{\sqrt{3}}$

Step 1.2.3.5.5

Simplify the numerator.

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Step 1.2.3.5.5.1

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

$x = \pm i \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 1.2.3.5.5.2

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

$x = \pm i \frac{2}{\sqrt{3}}$

Step 1.2.3.5.6

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.2.3.5.7

Combine and simplify the denominator.

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Step 1.2.3.5.7.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.3.5.7.2

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

$x = \pm i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.3.5.7.3

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

$x = \pm i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.3.5.7.4

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

$x = \pm i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.3.5.7.5

Add $1$ and $1$ .

$x = \pm i \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.3.5.7.6

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

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Step 1.2.3.5.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.2.3.5.7.6.2

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

$x = \pm i \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.5.7.6.3

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

$x = \pm i \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.5.7.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.5.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.5.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{2 \sqrt{3}}{3^{1}}$

Step 1.2.3.5.7.6.5

Evaluate the exponent.

$x = \pm i \frac{2 \sqrt{3}}{3}$

Step 1.2.3.5.8

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

$x = \pm \frac{i (2 \sqrt{3})}{3}$

Step 1.2.3.5.9

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

$x = \pm \frac{2 i \sqrt{3}}{3}$

Step 1.2.3.6

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

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Step 1.2.3.6.1

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

$x = \frac{2 i \sqrt{3}}{3}$

Step 1.2.3.6.2

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

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

Step 1.2.3.6.3

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

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

Step 2

Find the domain of $f (x) = \frac{x^{2} + 1}{x^{2} - 4}$ .

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Step 2.1

Set the denominator in $\frac{x^{2} + 1}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 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 2.2.3

Simplify $\pm \sqrt{4}$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

$x = \pm 2$

Step 2.2.4

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

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Step 2.2.4.1

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

$x = 2$

Step 2.2.4.2

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

$x = - 2$

Step 2.2.4.3

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

$x = 2, - 2$

Step 2.3

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Step 3

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

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 2)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply $3$ by $16$ .

$f'' (- 4) = \frac{10 (48 + 4)}{{(- 4 + 2)}^{3} {(- 4 - 2)}^{3}}$

Step 4.2.1.3

Add $48$ and $4$ .

$f'' (- 4) = \frac{10 \cdot 52}{{(- 4 + 2)}^{3} {(- 4 - 2)}^{3}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Add $- 4$ and $2$ .

$f'' (- 4) = \frac{10 \cdot 52}{{(- 2)}^{3} {(- 4 - 2)}^{3}}$

Step 4.2.2.2

Subtract $2$ from $- 4$ .

$f'' (- 4) = \frac{10 \cdot 52}{{(- 2)}^{3} {(- 6)}^{3}}$

Step 4.2.2.3

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

$f'' (- 4) = \frac{10 \cdot 52}{- 8 {(- 6)}^{3}}$

Step 4.2.2.4

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

$f'' (- 4) = \frac{10 \cdot 52}{- 8 \cdot - 216}$

Step 4.2.3

Reduce the expression by cancelling the common factors.

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Step 4.2.3.1

Multiply $10$ by $52$ .

$f'' (- 4) = \frac{520}{- 8 \cdot - 216}$

Step 4.2.3.2

Multiply $- 8$ by $- 216$ .

$f'' (- 4) = \frac{520}{1728}$

Step 4.2.3.3

Cancel the common factor of $520$ and $1728$ .

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Step 4.2.3.3.1

Factor $8$ out of $520$ .

$f'' (- 4) = \frac{8 (65)}{1728}$

Step 4.2.3.3.2

Cancel the common factors.

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Step 4.2.3.3.2.1

Factor $8$ out of $1728$ .

$f'' (- 4) = \frac{8 \cdot 65}{8 \cdot 216}$

Step 4.2.3.3.2.2

Cancel the common factor.

$f'' (- 4) = \frac{8 \cdot 65}{8 \cdot 216}$

Step 4.2.3.3.2.3

Rewrite the expression.

$f'' (- 4) = \frac{65}{216}$

Step 4.2.4

The final answer is $\frac{65}{216}$ .

$\frac{65}{216}$

Step 4.3

The graph is concave up on the interval $(- \infty, - 2)$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - 2)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 2, 2)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = \frac{10 (3 {(0)}^{2} + 4)}{{((0) + 2)}^{3} {((0) - 2)}^{3}}$

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

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

$f'' (0) = \frac{10 (3 \cdot 0 + 4)}{{(0 + 2)}^{3} {(0 - 2)}^{3}}$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = \frac{10 (0 + 4)}{{(0 + 2)}^{3} {(0 - 2)}^{3}}$

Step 5.2.1.3

Add $0$ and $4$ .

$f'' (0) = \frac{10 \cdot 4}{{(0 + 2)}^{3} {(0 - 2)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

$f'' (0) = \frac{10 \cdot 4}{{(- 1 \cdot 0 + 2)}^{3} {(0 - 2)}^{3}}$

Step 5.2.2.2

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

$f'' (0) = \frac{10 \cdot 4}{{(- 1 \cdot 0 - 1 \cdot - 2)}^{3} {(0 - 2)}^{3}}$

Step 5.2.2.3

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

$f'' (0) = \frac{10 \cdot 4}{{(- 1 \cdot (0 - 2))}^{3} {(0 - 2)}^{3}}$

Step 5.2.2.4

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

$f'' (0) = \frac{10 \cdot 4}{{(- 1)}^{3} \cdot ({(0 - 2)}^{3} {(0 - 2)}^{3})}$

Step 5.2.2.5

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

$f'' (0) = \frac{10 \cdot 4}{- 1 \cdot ({(0 - 2)}^{3} {(0 - 2)}^{3})}$

Step 5.2.2.6

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

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Step 5.2.2.6.1

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

$f'' (0) = \frac{10 \cdot 4}{- 1 \cdot ({(0 - 2)}^{3} {(0 - 2)}^{3})}$

Step 5.2.2.6.2

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

$f'' (0) = \frac{10 \cdot 4}{- 1 \cdot {(0 - 2)}^{3 + 3}}$

Step 5.2.2.6.3

Add $3$ and $3$ .

$f'' (0) = \frac{10 \cdot 4}{- 1 \cdot {(0 - 2)}^{6}}$

Step 5.2.3

Multiply $10$ by $4$ .

$f'' (0) = \frac{40}{- 1 \cdot {(0 - 2)}^{6}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Subtract $2$ from $0$ .

$f'' (0) = \frac{40}{- 1 {(- 2)}^{6}}$

Step 5.2.4.2

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

$f'' (0) = \frac{40}{- 1 \cdot 64}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Multiply $- 1$ by $64$ .

$f'' (0) = \frac{40}{- 64}$

Step 5.2.5.2

Cancel the common factor of $40$ and $- 64$ .

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Step 5.2.5.2.1

Factor $8$ out of $40$ .

$f'' (0) = \frac{8 (5)}{- 64}$

Step 5.2.5.2.2

Cancel the common factors.

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Step 5.2.5.2.2.1

Factor $8$ out of $- 64$ .

$f'' (0) = \frac{8 \cdot 5}{8 \cdot - 8}$

Step 5.2.5.2.2.2

Cancel the common factor.

$f'' (0) = \frac{8 \cdot 5}{8 \cdot - 8}$

Step 5.2.5.2.2.3

Rewrite the expression.

$f'' (0) = \frac{5}{- 8}$

Step 5.2.5.3

Move the negative in front of the fraction.

$f'' (0) = - \frac{5}{8}$

Step 5.2.6

The final answer is $- \frac{5}{8}$ .

$- \frac{5}{8}$

Step 5.3

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

Concave down on $(- 2, 2)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(2, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

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

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 $4$ to the power of $2$ .

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

Step 6.2.1.2

Multiply $3$ by $16$ .

$f'' (4) = \frac{10 (48 + 4)}{{(4 + 2)}^{3} {(4 - 2)}^{3}}$

Step 6.2.1.3

Add $48$ and $4$ .

$f'' (4) = \frac{10 \cdot 52}{{(4 + 2)}^{3} {(4 - 2)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $4$ and $2$ .

$f'' (4) = \frac{10 \cdot 52}{6^{3} {(4 - 2)}^{3}}$

Step 6.2.2.2

Subtract $2$ from $4$ .

$f'' (4) = \frac{10 \cdot 52}{6^{3} \cdot 2^{3}}$

Step 6.2.2.3

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

$f'' (4) = \frac{10 \cdot 52}{216 \cdot 2^{3}}$

Step 6.2.2.4

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

$f'' (4) = \frac{10 \cdot 52}{216 \cdot 8}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $10$ by $52$ .

$f'' (4) = \frac{520}{216 \cdot 8}$

Step 6.2.3.2

Multiply $216$ by $8$ .

$f'' (4) = \frac{520}{1728}$

Step 6.2.3.3

Cancel the common factor of $520$ and $1728$ .

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Step 6.2.3.3.1

Factor $8$ out of $520$ .

$f'' (4) = \frac{8 (65)}{1728}$

Step 6.2.3.3.2

Cancel the common factors.

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Step 6.2.3.3.2.1

Factor $8$ out of $1728$ .

$f'' (4) = \frac{8 \cdot 65}{8 \cdot 216}$

Step 6.2.3.3.2.2

Cancel the common factor.

$f'' (4) = \frac{8 \cdot 65}{8 \cdot 216}$

Step 6.2.3.3.2.3

Rewrite the expression.

$f'' (4) = \frac{65}{216}$

Step 6.2.4

The final answer is $\frac{65}{216}$ .

$\frac{65}{216}$

Step 6.3

The graph is concave up on the interval $(2, \infty)$ because $f'' (4)$ is positive.

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 7

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

Concave up on $(- \infty, - 2)$ since $f'' (x)$ is positive

Concave down on $(- 2, 2)$ since $f'' (x)$ is negative

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 8