Calculus Examples

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

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

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.1.1.2.6

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

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

Step 1.1.1.2.7

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

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

Step 1.1.1.2.8

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

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

Step 1.1.1.2.9

Simplify the expression.

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Step 1.1.1.2.9.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.9.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.3.3

Simplify the numerator.

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Step 1.1.1.3.3.1

Simplify each term.

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Step 1.1.1.3.3.1.1

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

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Step 1.1.1.3.3.1.1.1

Apply the distributive property.

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

Step 1.1.1.3.3.1.1.2

Apply the distributive property.

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

Step 1.1.1.3.3.1.1.3

Apply the distributive property.

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

Step 1.1.1.3.3.1.2

Simplify each term.

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Step 1.1.1.3.3.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.1.3.3.1.2.2

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

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Step 1.1.1.3.3.1.2.2.1

Move $x$ .

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

Step 1.1.1.3.3.1.2.2.2

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

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Step 1.1.1.3.3.1.2.2.2.1

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

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

Step 1.1.1.3.3.1.2.2.2.2

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

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

Step 1.1.1.3.3.1.2.2.3

Add $1$ and $2$ .

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

Step 1.1.1.3.3.1.2.3

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

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

Step 1.1.1.3.3.1.2.4

Multiply $2$ by $- 4$ .

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

Step 1.1.1.3.3.1.2.5

Multiply $- 4$ by $- 2$ .

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

Step 1.1.1.3.3.1.3

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

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Step 1.1.1.3.3.1.3.1

Move $x$ .

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

Step 1.1.1.3.3.1.3.2

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

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Step 1.1.1.3.3.1.3.2.1

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

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

Step 1.1.1.3.3.1.3.2.2

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

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

Step 1.1.1.3.3.1.3.3

Add $1$ and $2$ .

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

Step 1.1.1.3.3.1.4

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

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Step 1.1.1.3.3.1.4.1

Move $x$ .

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

Step 1.1.1.3.3.1.4.2

Multiply $x$ by $x$ .

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

Step 1.1.1.3.3.1.5

Multiply $- 2$ by $- 2$ .

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

Step 1.1.1.3.3.2

Combine the opposite terms in $2 x^{3} - 2 x^{2} - 8 x + 8 - 2 x^{3} + 4 x^{2}$ .

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Step 1.1.1.3.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{- 2 x^{2} - 8 x + 8 + 0 + 4 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.1.1.3.3.2.2

Add $- 2 x^{2} - 8 x + 8$ and $0$ .

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

Step 1.1.1.3.3.3

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

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

Step 1.1.1.3.4

Simplify the numerator.

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Step 1.1.1.3.4.1

Factor $2$ out of $2 x^{2} - 8 x + 8$ .

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Step 1.1.1.3.4.1.1

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

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

Step 1.1.1.3.4.1.2

Factor $2$ out of $- 8 x$ .

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

Step 1.1.1.3.4.1.3

Factor $2$ out of $8$ .

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

Step 1.1.1.3.4.1.4

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

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

Step 1.1.1.3.4.1.5

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

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

Step 1.1.1.3.4.2

Factor using the perfect square rule.

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Step 1.1.1.3.4.2.1

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

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

Step 1.1.1.3.4.2.2

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

$4 x = 2 \cdot x \cdot 2$

Step 1.1.1.3.4.2.3

Rewrite the polynomial.

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

Step 1.1.1.3.4.2.4

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

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

Step 1.1.1.3.5

Simplify the denominator.

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Step 1.1.1.3.5.1

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

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

Step 1.1.1.3.5.2

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 $b = 2$ .

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

Step 1.1.1.3.5.3

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

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

Step 1.1.1.3.6

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

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Step 1.1.1.3.6.1

Cancel the common factor.

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

Step 1.1.1.3.6.2

Rewrite the expression.

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Apply basic rules of exponents.

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Step 1.1.2.1.2.1

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

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

Step 1.1.2.1.2.2

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

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Step 1.1.2.1.2.2.1

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

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

Step 1.1.2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $x + 2$ .

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

Replace all occurrences of $u$ with $x + 2$ .

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

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

Multiply $- 2$ by $2$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

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

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

Step 1.1.2.3.4

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

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

Step 1.1.2.3.5

Simplify the expression.

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Step 1.1.2.3.5.1

Add $1$ and $0$ .

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

Step 1.1.2.3.5.2

Multiply $- 4$ by $1$ .

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

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

Combine terms.

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Step 1.1.2.4.2.1

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

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

Step 1.1.2.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

Set the second derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$4 = 0$

Step 1.2.3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 2

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

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Step 2.1

Set the denominator in $\frac{x^{2} - 2 x}{x^{2} - 4}$ 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.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the denominator.

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Step 4.2.1.1

Add $- 4$ and $2$ .

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

Step 4.2.1.2

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

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

Step 4.2.2

Reduce the expression by cancelling the common factors.

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Step 4.2.2.1

Cancel the common factor of $4$ and $- 8$ .

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Step 4.2.2.1.1

Factor $4$ out of $4$ .

$f'' (- 4) = - \frac{4 (1)}{- 8}$

Step 4.2.2.1.2

Cancel the common factors.

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Step 4.2.2.1.2.1

Factor $4$ out of $- 8$ .

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

Step 4.2.2.1.2.2

Cancel the common factor.

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

Step 4.2.2.1.2.3

Rewrite the expression.

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

Step 4.2.2.2

Move the negative in front of the fraction.

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

Step 4.2.3

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

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.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the denominator.

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Step 5.2.1.1

Add $0$ and $2$ .

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

Step 5.2.1.2

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

$f'' (0) = - \frac{4}{8}$

Step 5.2.2

Cancel the common factor of $4$ and $8$ .

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Step 5.2.2.1

Factor $4$ out of $4$ .

$f'' (0) = - \frac{4 (1)}{8}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $4$ out of $8$ .

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

Step 5.2.2.2.2

Cancel the common factor.

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

Step 5.2.2.2.3

Rewrite the expression.

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

Step 5.2.3

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

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.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Add $4$ and $2$ .

$f'' (4) = - \frac{4}{6^{3}}$

Step 6.2.1.2

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

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

Step 6.2.2

Cancel the common factor of $4$ and $216$ .

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Step 6.2.2.1

Factor $4$ out of $4$ .

$f'' (4) = - \frac{4 (1)}{216}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $4$ out of $216$ .

$f'' (4) = - \frac{4 \cdot 1}{4 \cdot 54}$

Step 6.2.2.2.2

Cancel the common factor.

$f'' (4) = - \frac{4 \cdot 1}{4 \cdot 54}$

Step 6.2.2.2.3

Rewrite the expression.

$f'' (4) = - \frac{1}{54}$

Step 6.2.3

The final answer is $- \frac{1}{54}$ .

$- \frac{1}{54}$

Step 6.3

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

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

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 down on $(2, \infty)$ since $f'' (x)$ is negative

Step 8