Calculus Examples

$f (x) = \frac{1}{x^{2} - 6 x - 7}$

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

Rewrite $\frac{1}{x^{2} - 6 x - 7}$ as ${(x^{2} - 6 x - 7)}^{- 1}$ .

$f' (x) = \frac{d}{d x} ({(x^{2} - 6 x - 7)}^{- 1})$

Step 1.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 6 x - 7$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 6 x - 7$ .

$f' (x) = \frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2} - 6 x - 7)$

Step 1.1.1.2.2

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

$f' (x) = - u^{- 2} \frac{d}{d x} (x^{2} - 6 x - 7)$

Step 1.1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 6 x - 7$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} \frac{d}{d x} (x^{2} - 6 x - 7)$

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

By the Sum Rule, the derivative of $x^{2} - 6 x - 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 7]$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6 x) + \frac{d}{d x} (- 7))$

Step 1.1.1.3.2

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

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x + \frac{d}{d x} (- 6 x) + \frac{d}{d x} (- 7))$

Step 1.1.1.3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 \frac{d}{d x} x + \frac{d}{d x} (- 7))$

Step 1.1.1.3.4

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

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 \cdot 1 + \frac{d}{d x} (- 7))$

Step 1.1.1.3.5

Multiply $- 6$ by $1$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 + \frac{d}{d x} (- 7))$

Step 1.1.1.3.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6 + 0)$

Step 1.1.1.3.7

Add $2 x - 6$ and $0$ .

$f' (x) = - {(x^{2} - 6 x - 7)}^{- 2} (2 x - 6)$

Step 1.1.1.4

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

$f' (x) = - \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (2 x - 6)$

Step 1.1.1.5

Reorder the factors of $- \frac{1}{{(x^{2} - 6 x - 7)}^{2}} (2 x - 6)$ .

$f' (x) = - (2 x - 6) \frac{1}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (2 x - 6) \frac{1}{{(x^{2} - 6 x - 7)}^{2}}$ with respect to $x$ is $- \frac{d}{d x} [(2 x - 6) \frac{1}{{(x^{2} - 6 x - 7)}^{2}}]$ .

$f'' (x) = - \frac{d}{d x} ((2 x - 6) (\frac{1}{{(x^{2} - 6 x - 7)}^{2}}))$

Step 1.1.2.2

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 $f (x) = 2 x - 6$ and $g (x) = \frac{1}{{(x^{2} - 6 x - 7)}^{2}}$ .

$f'' (x) = - ((2 x - 6) \frac{d}{d x} (\frac{1}{{(x^{2} - 6 x - 7)}^{2}}) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.3

Apply basic rules of exponents.

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Step 1.1.2.3.1

Rewrite $\frac{1}{{(x^{2} - 6 x - 7)}^{2}}$ as ${({(x^{2} - 6 x - 7)}^{2})}^{- 1}$ .

$f'' (x) = - ((2 x - 6) \frac{d}{d x} ({({(x^{2} - 6 x - 7)}^{2})}^{- 1}) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.3.2

Multiply the exponents in ${({(x^{2} - 6 x - 7)}^{2})}^{- 1}$ .

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Step 1.1.2.3.2.1

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

$f'' (x) = - ((2 x - 6) \frac{d}{d x} ({(x^{2} - 6 x - 7)}^{2 \cdot - 1}) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.3.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - ((2 x - 6) \frac{d}{d x} ({(x^{2} - 6 x - 7)}^{- 2}) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.4

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 $g (x) = x^{2} - 6 x - 7$ .

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Step 1.1.2.4.1

To apply the Chain Rule, set $u$ as $x^{2} - 6 x - 7$ .

$f'' (x) = - ((2 x - 6) (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (x^{2} - 6 x - 7)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.4.2

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

$f'' (x) = - ((2 x - 6) (- 2 u^{- 3} \frac{d}{d x} (x^{2} - 6 x - 7)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.4.3

Replace all occurrences of $u$ with $x^{2} - 6 x - 7$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} \frac{d}{d x} (x^{2} - 6 x - 7)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5

Differentiate.

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Step 1.1.2.5.1

By the Sum Rule, the derivative of $x^{2} - 6 x - 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 7]$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6 x) + \frac{d}{d x} (- 7))) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.2

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

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x + \frac{d}{d x} (- 6 x) + \frac{d}{d x} (- 7))) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x - 6 \frac{d}{d x} x + \frac{d}{d x} (- 7))) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.4

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

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x - 6 \cdot 1 + \frac{d}{d x} (- 7))) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.5

Multiply $- 6$ by $1$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x - 6 + \frac{d}{d x} (- 7))) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x - 6 + 0)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.5.7

Add $2 x - 6$ and $0$ .

$f'' (x) = - ((2 x - 6) (- 2 {(x^{2} - 6 x - 7)}^{- 3} (2 x - 6)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.6

Raise $2 x - 6$ to the power of $1$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} ((2 x - 6) (2 x - 6)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.7

Raise $2 x - 6$ to the power of $1$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} ((2 x - 6) (2 x - 6)) + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.8

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

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{1 + 1} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.9

Add $1$ and $1$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot \frac{d}{d x} (2 x - 6))$

Step 1.1.2.10

By the Sum Rule, the derivative of $2 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (\frac{d}{d x} (2 x) + \frac{d}{d x} (- 6)))$

Step 1.1.2.11

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

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (2 \frac{d}{d x} (x) + \frac{d}{d x} (- 6)))$

Step 1.1.2.12

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

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (2 \cdot 1 + \frac{d}{d x} (- 6)))$

Step 1.1.2.13

Multiply $2$ by $1$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (2 + \frac{d}{d x} (- 6)))$

Step 1.1.2.14

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot (2 + 0))$

Step 1.1.2.15

Combine fractions.

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Step 1.1.2.15.1

Add $2$ and $0$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{1}{{(x^{2} - 6 x - 7)}^{2}} \cdot 2)$

Step 1.1.2.15.2

Combine $\frac{1}{{(x^{2} - 6 x - 7)}^{2}}$ and $2$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} + \frac{2}{{(x^{2} - 6 x - 7)}^{2}})$

Step 1.1.2.16

To write $- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - 6 x - 7)}^{2}}{{(x^{2} - 6 x - 7)}^{2}}$ .

$f'' (x) = - (- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} \cdot \frac{{(x^{2} - 6 x - 7)}^{2}}{{(x^{2} - 6 x - 7)}^{2}} + \frac{2}{{(x^{2} - 6 x - 7)}^{2}})$

Step 1.1.2.17

Combine $- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2}$ and $\frac{{(x^{2} - 6 x - 7)}^{2}}{{(x^{2} - 6 x - 7)}^{2}}$ .

$f'' (x) = - (\frac{- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} {(x^{2} - 6 x - 7)}^{2}}{{(x^{2} - 6 x - 7)}^{2}} + \frac{2}{{(x^{2} - 6 x - 7)}^{2}})$

Step 1.1.2.18

Combine the numerators over the common denominator.

$f'' (x) = - \frac{- 2 {(x^{2} - 6 x - 7)}^{- 3} {(2 x - 6)}^{2} {(x^{2} - 6 x - 7)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.19

Multiply ${(x^{2} - 6 x - 7)}^{- 3}$ by ${(x^{2} - 6 x - 7)}^{2}$ by adding the exponents.

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Step 1.1.2.19.1

Move ${(x^{2} - 6 x - 7)}^{2}$ .

$f'' (x) = - \frac{- 2 ({(x^{2} - 6 x - 7)}^{2} {(x^{2} - 6 x - 7)}^{- 3}) {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.19.2

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

$f'' (x) = - \frac{- 2 {(x^{2} - 6 x - 7)}^{2 - 3} {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.19.3

Subtract $3$ from $2$ .

$f'' (x) = - \frac{- 2 {(x^{2} - 6 x - 7)}^{- 1} {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20

Simplify.

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Step 1.1.2.20.1

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

$f'' (x) = - \frac{- 2 \frac{1}{x^{2} - 6 x - 7} \cdot {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2

Simplify each term.

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Step 1.1.2.20.2.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 1.1.2.20.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 1.1.2.20.2.1.2

Write the factored form using these integers.

$f'' (x) = - \frac{- 2 \frac{1}{(x - 7) (x + 1)} \cdot {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.2

Combine $- 2$ and $\frac{1}{(x - 7) (x + 1)}$ .

$f'' (x) = - \frac{\frac{- 2}{(x - 7) (x + 1)} \cdot {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot {(2 x - 6)}^{2} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.4

Rewrite ${(2 x - 6)}^{2}$ as $(2 x - 6) (2 x - 6)$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot ((2 x - 6) (2 x - 6)) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.5

Expand $(2 x - 6) (2 x - 6)$ using the FOIL Method.

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Step 1.1.2.20.2.5.1

Apply the distributive property.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 x (2 x - 6) - 6 (2 x - 6)) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.5.2

Apply the distributive property.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 x (2 x) + 2 x \cdot - 6 - 6 (2 x - 6)) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.5.3

Apply the distributive property.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 x (2 x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6

Simplify and combine like terms.

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Step 1.1.2.20.2.6.1

Simplify each term.

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Step 1.1.2.20.2.6.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 \cdot (2 x \cdot x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.2

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

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Step 1.1.2.20.2.6.1.2.1

Move $x$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (2 \cdot (2 x^{2}) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.3

Multiply $2$ by $2$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2} + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.4

Multiply $- 6$ by $2$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2} - 12 x - 6 (2 x) - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.5

Multiply $2$ by $- 6$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2} - 12 x - 12 x - 6 \cdot - 6) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.1.6

Multiply $- 6$ by $- 6$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2} - 12 x - 12 x + 36) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.6.2

Subtract $12 x$ from $- 12 x$ .

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2} - 24 x + 36) + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.7

Apply the distributive property.

$f'' (x) = - \frac{- \frac{2}{(x - 7) (x + 1)} \cdot (4 x^{2}) - \frac{2}{(x - 7) (x + 1)} \cdot (- 24 x) - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8

Simplify.

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Step 1.1.2.20.2.8.1

Multiply $- \frac{2}{(x - 7) (x + 1)} (4 x^{2})$ .

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Step 1.1.2.20.2.8.1.1

Multiply $4$ by $- 1$ .

$f'' (x) = - \frac{- 4 \frac{2}{(x - 7) (x + 1)} \cdot x^{2} - \frac{2}{(x - 7) (x + 1)} \cdot (- 24 x) - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.1.2

Combine $- 4$ and $\frac{2}{(x - 7) (x + 1)}$ .

$f'' (x) = - \frac{\frac{- 4 \cdot 2}{(x - 7) (x + 1)} \cdot x^{2} - \frac{2}{(x - 7) (x + 1)} \cdot (- 24 x) - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.1.3

Multiply $- 4$ by $2$ .

$f'' (x) = - \frac{\frac{- 8}{(x - 7) (x + 1)} \cdot x^{2} - \frac{2}{(x - 7) (x + 1)} \cdot (- 24 x) - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.1.4

Combine $\frac{- 8}{(x - 7) (x + 1)}$ and $x^{2}$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} - \frac{2}{(x - 7) (x + 1)} \cdot (- 24 x) - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.2

Multiply $- \frac{2}{(x - 7) (x + 1)} (- 24 x)$ .

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Step 1.1.2.20.2.8.2.1

Multiply $- 24$ by $- 1$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + 24 (\frac{2}{(x - 7) (x + 1)}) \cdot x - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.2.2

Combine $24$ and $\frac{2}{(x - 7) (x + 1)}$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{24 \cdot 2}{(x - 7) (x + 1)} \cdot x - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.2.3

Multiply $24$ by $2$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{48}{(x - 7) (x + 1)} \cdot x - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.2.4

Combine $\frac{48}{(x - 7) (x + 1)}$ and $x$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - \frac{2}{(x - 7) (x + 1)} \cdot 36 + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.3

Multiply $- \frac{2}{(x - 7) (x + 1)} \cdot 36$ .

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Step 1.1.2.20.2.8.3.1

Multiply $36$ by $- 1$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - 36 \frac{2}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.3.2

Combine $- 36$ and $\frac{2}{(x - 7) (x + 1)}$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} + \frac{- 36 \cdot 2}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.8.3.3

Multiply $- 36$ by $2$ .

$f'' (x) = - \frac{\frac{- 8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} + \frac{- 72}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.9

Simplify each term.

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Step 1.1.2.20.2.9.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{(8) x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} + \frac{- 72}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.2.9.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - \frac{72}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.3

Combine terms.

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Step 1.1.2.20.3.1

Multiply $\frac{- \frac{8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - \frac{72}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}}$ by $\frac{(x - 7) (x + 1)}{(x - 7) (x + 1)}$ .

$f'' (x) = - (\frac{(x - 7) (x + 1)}{(x - 7) (x + 1)} \cdot \frac{- \frac{8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - \frac{72}{(x - 7) (x + 1)} + 2}{{(x^{2} - 6 x - 7)}^{2}})$

Step 1.1.2.20.3.2

Combine.

$f'' (x) = - \frac{(x - 7) (x + 1) (- \frac{8 x^{2}}{(x - 7) (x + 1)} + \frac{48 x}{(x - 7) (x + 1)} - \frac{72}{(x - 7) (x + 1)} + 2)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.3.3

Apply the distributive property.

$f'' (x) = - \frac{(x - 7) (x + 1) (- \frac{8 x^{2}}{(x - 7) (x + 1)}) + (x - 7) (x + 1) (\frac{48 x}{(x - 7) (x + 1)}) + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + (x - 7) (x + 1) \cdot 2}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.3.4

Cancel the common factor of $(x - 7) (x + 1)$ .

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Step 1.1.2.20.3.4.1

Cancel the common factor.

Step 1.1.2.20.3.4.2

Rewrite the expression.

$f'' (x) = - \frac{(x - 7) (x + 1) (- \frac{8 x^{2}}{(x - 7) (x + 1)}) + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + (x - 7) (x + 1) \cdot 2}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.3.5

Move $2$ to the left of $(x - 7) (x + 1)$ .

$f'' (x) = - \frac{(x - 7) (x + 1) (- \frac{8 x^{2}}{(x - 7) (x + 1)}) + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4

Simplify the numerator.

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Step 1.1.2.20.4.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- (x - 7) (x + 1) \frac{8 x^{2}}{(x - 7) (x + 1)} + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.2

Cancel the common factor of $(x - 7) (x + 1)$ .

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Step 1.1.2.20.4.2.1

Factor $(x - 7) (x + 1)$ out of $- (x - 7) (x + 1)$ .

$f'' (x) = - \frac{(x - 7) (x + 1) (- 1) (\frac{8 x^{2}}{(x - 7) (x + 1)}) + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x - 7) (x + 1) \cdot (- 1 \frac{8 x^{2}}{(x - 7) (x + 1)}) + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.2.3

Rewrite the expression.

$f'' (x) = - \frac{- (8 x^{2}) + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.3

Multiply $8$ by $- 1$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x + (x - 7) (x + 1) (- \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - (x - 7) (x + 1) \frac{72}{(x - 7) (x + 1)} + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.5

Cancel the common factor of $(x - 7) (x + 1)$ .

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Step 1.1.2.20.4.5.1

Factor $(x - 7) (x + 1)$ out of $- (x - 7) (x + 1)$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x + (x - 7) (x + 1) (- 1) (\frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.5.2

Cancel the common factor.

$f'' (x) = - \frac{- 8 x^{2} + 48 x + (x - 7) (x + 1) \cdot (- 1 \frac{72}{(x - 7) (x + 1)}) + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.5.3

Rewrite the expression.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 1 \cdot 72 + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.6

Multiply $- 1$ by $72$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 (x - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.7

Apply the distributive property.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + (2 x + 2 \cdot - 7) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.8

Multiply $2$ by $- 7$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + (2 x - 14) (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.9

Expand $(2 x - 14) (x + 1)$ using the FOIL Method.

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Step 1.1.2.20.4.9.1

Apply the distributive property.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x (x + 1) - 14 (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.9.2

Apply the distributive property.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x \cdot x + 2 x \cdot 1 - 14 (x + 1)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.9.3

Apply the distributive property.

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x \cdot x + 2 x \cdot 1 - 14 x - 14 \cdot 1}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.10

Simplify and combine like terms.

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Step 1.1.2.20.4.10.1

Simplify each term.

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Step 1.1.2.20.4.10.1.1

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

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Step 1.1.2.20.4.10.1.1.1

Move $x$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 (x \cdot x) + 2 x \cdot 1 - 14 x - 14 \cdot 1}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.10.1.1.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x^{2} + 2 x \cdot 1 - 14 x - 14 \cdot 1}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.10.1.2

Multiply $2$ by $1$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x^{2} + 2 x - 14 x - 14 \cdot 1}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.10.1.3

Multiply $- 14$ by $1$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x^{2} + 2 x - 14 x - 14}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.10.2

Subtract $14 x$ from $2 x$ .

$f'' (x) = - \frac{- 8 x^{2} + 48 x - 72 + 2 x^{2} - 12 x - 14}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.11

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

$f'' (x) = - \frac{- 6 x^{2} + 48 x - 72 - 12 x - 14}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.12

Subtract $12 x$ from $48 x$ .

$f'' (x) = - \frac{- 6 x^{2} + 36 x - 72 - 14}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.13

Subtract $14$ from $- 72$ .

$f'' (x) = - \frac{- 6 x^{2} + 36 x - 86}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.14

Factor $2$ out of $- 6 x^{2} + 36 x - 86$ .

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Step 1.1.2.20.4.14.1

Factor $2$ out of $- 6 x^{2}$ .

$f'' (x) = - \frac{2 (- 3 x^{2}) + 36 x - 86}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.14.2

Factor $2$ out of $36 x$ .

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2 (18 x) - 86}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.14.3

Factor $2$ out of $- 86$ .

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2 (18 x) + 2 (- 43)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.14.4

Factor $2$ out of $2 (- 3 x^{2}) + 2 (18 x)$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x) + 2 (- 43)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.4.14.5

Factor $2$ out of $2 (- 3 x^{2} + 18 x) + 2 (- 43)$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{(x - 7) (x + 1) {(x^{2} - 6 x - 7)}^{2}}$

Step 1.1.2.20.5

Simplify the denominator.

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Step 1.1.2.20.5.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 1.1.2.20.5.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 1.1.2.20.5.1.2

Write the factored form using these integers.

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{(x - 7) (x + 1) {((x - 7) (x + 1))}^{2}}$

Step 1.1.2.20.5.2

Apply the product rule to $(x - 7) (x + 1)$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{(x - 7) (x + 1) {(x - 7)}^{2} {(x + 1)}^{2}}$

Step 1.1.2.20.5.3

Combine exponents.

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Step 1.1.2.20.5.3.1

Multiply $x - 7$ by ${(x - 7)}^{2}$ by adding the exponents.

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Step 1.1.2.20.5.3.1.1

Move ${(x - 7)}^{2}$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{2} (x - 7) (x + 1) {(x + 1)}^{2}}$

Step 1.1.2.20.5.3.1.2

Multiply ${(x - 7)}^{2}$ by $x - 7$ .

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Step 1.1.2.20.5.3.1.2.1

Raise $x - 7$ to the power of $1$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{2} (x - 7) (x + 1) {(x + 1)}^{2}}$

Step 1.1.2.20.5.3.1.2.2

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

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{2 + 1} (x + 1) {(x + 1)}^{2}}$

Step 1.1.2.20.5.3.1.3

Add $2$ and $1$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{3} (x + 1) {(x + 1)}^{2}}$

Step 1.1.2.20.5.3.2

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

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Step 1.1.2.20.5.3.2.1

Move ${(x + 1)}^{2}$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{3} ({(x + 1)}^{2} (x + 1))}$

Step 1.1.2.20.5.3.2.2

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

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Step 1.1.2.20.5.3.2.2.1

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

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{3} ({(x + 1)}^{2} (x + 1))}$

Step 1.1.2.20.5.3.2.2.2

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

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{3} {(x + 1)}^{2 + 1}}$

Step 1.1.2.20.5.3.2.3

Add $2$ and $1$ .

$f'' (x) = - \frac{2 (- 3 x^{2} + 18 x - 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.6

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

$f'' (x) = - \frac{2 (- (3 x^{2}) + 18 x - 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.7

Factor $- 1$ out of $18 x$ .

$f'' (x) = - \frac{2 (- (3 x^{2}) - (- 18 x) - 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.8

Factor $- 1$ out of $- (3 x^{2}) - (- 18 x)$ .

$f'' (x) = - \frac{2 (- (3 x^{2} - 18 x) - 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.9

Rewrite $- 43$ as $- 1 (43)$ .

$f'' (x) = - \frac{2 (- (3 x^{2} - 18 x) - 1 \cdot 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.10

Factor $- 1$ out of $- (3 x^{2} - 18 x) - 1 (43)$ .

$f'' (x) = - \frac{2 (- (3 x^{2} - 18 x + 43))}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.11

Rewrite $- (3 x^{2} - 18 x + 43)$ as $- 1 (3 x^{2} - 18 x + 43)$ .

$f'' (x) = - \frac{2 (- 1 (3 x^{2} - 18 x + 43))}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.12

Move the negative in front of the fraction.

$f'' (x) = \frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.2.20.13

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}})$

Step 1.1.2.20.14

Multiply $\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$ .

$\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{2 (3 x^{2} - 18 x + 43)}{{(x - 7)}^{3} {(x + 1)}^{3}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 (3 x^{2} - 18 x + 43) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $2 (3 x^{2} - 18 x + 43) = 0$ by $2$ and simplify.

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Step 1.2.3.1.1

Divide each term in $2 (3 x^{2} - 18 x + 43) = 0$ by $2$ .

$\frac{2 (3 x^{2} - 18 x + 43)}{2} = \frac{0}{2}$

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 $2$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (3 x^{2} - 18 x + 43)}{2} = \frac{0}{2}$

Step 1.2.3.1.2.1.2

Divide $3 x^{2} - 18 x + 43$ by $1$ .

$3 x^{2} - 18 x + 43 = \frac{0}{2}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $2$ .

$3 x^{2} - 18 x + 43 = 0$

Step 1.2.3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.3.3

Substitute the values $a = 3$ , $b = - 18$ , and $c = 43$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (3 \cdot 43)}}{2 \cdot 3}$

Step 1.2.3.4

Simplify.

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Step 1.2.3.4.1

Simplify the numerator.

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Step 1.2.3.4.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 3 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.4.1.2

Multiply $- 4 \cdot 3 \cdot 43$ .

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Step 1.2.3.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{18 \pm \sqrt{324 - 12 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.4.1.2.2

Multiply $- 12$ by $43$ .

$x = \frac{18 \pm \sqrt{324 - 516}}{2 \cdot 3}$

Step 1.2.3.4.1.3

Subtract $516$ from $324$ .

$x = \frac{18 \pm \sqrt{- 192}}{2 \cdot 3}$

Step 1.2.3.4.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{18 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 3}$

Step 1.2.3.4.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{18 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.4.1.6

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

$x = \frac{18 \pm i \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.4.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 1.2.3.4.1.7.1

Factor $64$ out of $192$ .

$x = \frac{18 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 3}$

Step 1.2.3.4.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{18 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 3}$

Step 1.2.3.4.1.8

Pull terms out from under the radical.

$x = \frac{18 \pm i \cdot (8 \sqrt{3})}{2 \cdot 3}$

Step 1.2.3.4.1.9

Move $8$ to the left of $i$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{2 \cdot 3}$

Step 1.2.3.4.2

Multiply $2$ by $3$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{6}$

Step 1.2.3.4.3

Simplify $\frac{18 \pm 8 i \sqrt{3}}{6}$ .

$x = \frac{9 \pm 4 i \sqrt{3}}{3}$

Step 1.2.3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.3.5.1

Simplify the numerator.

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Step 1.2.3.5.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 3 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.5.1.2

Multiply $- 4 \cdot 3 \cdot 43$ .

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Step 1.2.3.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{18 \pm \sqrt{324 - 12 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.5.1.2.2

Multiply $- 12$ by $43$ .

$x = \frac{18 \pm \sqrt{324 - 516}}{2 \cdot 3}$

Step 1.2.3.5.1.3

Subtract $516$ from $324$ .

$x = \frac{18 \pm \sqrt{- 192}}{2 \cdot 3}$

Step 1.2.3.5.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{18 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 3}$

Step 1.2.3.5.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{18 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.5.1.6

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

$x = \frac{18 \pm i \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.5.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 1.2.3.5.1.7.1

Factor $64$ out of $192$ .

$x = \frac{18 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 3}$

Step 1.2.3.5.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{18 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 3}$

Step 1.2.3.5.1.8

Pull terms out from under the radical.

$x = \frac{18 \pm i \cdot (8 \sqrt{3})}{2 \cdot 3}$

Step 1.2.3.5.1.9

Move $8$ to the left of $i$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{2 \cdot 3}$

Step 1.2.3.5.2

Multiply $2$ by $3$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{6}$

Step 1.2.3.5.3

Simplify $\frac{18 \pm 8 i \sqrt{3}}{6}$ .

$x = \frac{9 \pm 4 i \sqrt{3}}{3}$

Step 1.2.3.5.4

Change the $\pm$ to $+$ .

$x = \frac{9 + 4 i \sqrt{3}}{3}$

Step 1.2.3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.3.6.1

Simplify the numerator.

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Step 1.2.3.6.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 3 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.6.1.2

Multiply $- 4 \cdot 3 \cdot 43$ .

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Step 1.2.3.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{18 \pm \sqrt{324 - 12 \cdot 43}}{2 \cdot 3}$

Step 1.2.3.6.1.2.2

Multiply $- 12$ by $43$ .

$x = \frac{18 \pm \sqrt{324 - 516}}{2 \cdot 3}$

Step 1.2.3.6.1.3

Subtract $516$ from $324$ .

$x = \frac{18 \pm \sqrt{- 192}}{2 \cdot 3}$

Step 1.2.3.6.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{18 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 3}$

Step 1.2.3.6.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{18 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.6.1.6

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

$x = \frac{18 \pm i \cdot \sqrt{192}}{2 \cdot 3}$

Step 1.2.3.6.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 1.2.3.6.1.7.1

Factor $64$ out of $192$ .

$x = \frac{18 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 3}$

Step 1.2.3.6.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{18 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 3}$

Step 1.2.3.6.1.8

Pull terms out from under the radical.

$x = \frac{18 \pm i \cdot (8 \sqrt{3})}{2 \cdot 3}$

Step 1.2.3.6.1.9

Move $8$ to the left of $i$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{2 \cdot 3}$

Step 1.2.3.6.2

Multiply $2$ by $3$ .

$x = \frac{18 \pm 8 i \sqrt{3}}{6}$

Step 1.2.3.6.3

Simplify $\frac{18 \pm 8 i \sqrt{3}}{6}$ .

$x = \frac{9 \pm 4 i \sqrt{3}}{3}$

Step 1.2.3.6.4

Change the $\pm$ to $-$ .

$x = \frac{9 - 4 i \sqrt{3}}{3}$

Step 1.2.3.7

The final answer is the combination of both solutions.

$x = \frac{9 + 4 i \sqrt{3}}{3}, \frac{9 - 4 i \sqrt{3}}{3}$

Step 2

Find the domain of $f (x) = \frac{1}{x^{2} - 6 x - 7}$ .

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Step 2.1

Set the denominator in $\frac{1}{x^{2} - 6 x - 7}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 6 x - 7 = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 2.2.1.2

Write the factored form using these integers.

$(x - 7) (x + 1) = 0$

Step 2.2.2

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

$x - 7 = 0$

$x + 1 = 0$

Step 2.2.3

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 2.2.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.5

The final solution is all the values that make $(x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 2.3

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 7}$

Interval Notation:

$(- \infty, - 1) \cup (- 1, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 7}$

Step 3

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

$(- \infty, - 1) \cup (- 1, 7) \cup (7, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 1)$ 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.

$f'' (- 4) = \frac{2 (3 {(- 4)}^{2} - 18 \cdot - 4 + 43)}{{((- 4) - 7)}^{3} {((- 4) + 1)}^{3}}$

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 $2$ .

$f'' (- 4) = \frac{2 (3 \cdot 16 - 18 \cdot - 4 + 43)}{{(- 4 - 7)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.1.2

Multiply $3$ by $16$ .

$f'' (- 4) = \frac{2 (48 - 18 \cdot - 4 + 43)}{{(- 4 - 7)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.1.3

Multiply $- 18$ by $- 4$ .

$f'' (- 4) = \frac{2 (48 + 72 + 43)}{{(- 4 - 7)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.1.4

Add $48$ and $72$ .

$f'' (- 4) = \frac{2 (120 + 43)}{{(- 4 - 7)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.1.5

Add $120$ and $43$ .

$f'' (- 4) = \frac{2 \cdot 163}{{(- 4 - 7)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Subtract $7$ from $- 4$ .

$f'' (- 4) = \frac{2 \cdot 163}{{(- 11)}^{3} {(- 4 + 1)}^{3}}$

Step 4.2.2.2

Add $- 4$ and $1$ .

$f'' (- 4) = \frac{2 \cdot 163}{{(- 11)}^{3} {(- 3)}^{3}}$

Step 4.2.2.3

Raise $- 11$ to the power of $3$ .

$f'' (- 4) = \frac{2 \cdot 163}{- 1331 {(- 3)}^{3}}$

Step 4.2.2.4

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

$f'' (- 4) = \frac{2 \cdot 163}{- 1331 \cdot - 27}$

Step 4.2.3

Multiply.

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Step 4.2.3.1

Multiply $2$ by $163$ .

$f'' (- 4) = \frac{326}{- 1331 \cdot - 27}$

Step 4.2.3.2

Multiply $- 1331$ by $- 27$ .

$f'' (- 4) = \frac{326}{35937}$

Step 4.2.4

The final answer is $\frac{326}{35937}$ .

$\frac{326}{35937}$

Step 4.3

The graph is concave up on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 1, 7)$ 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.

$f'' (0) = \frac{2 (3 {(0)}^{2} - 18 \cdot 0 + 43)}{{((0) - 7)}^{3} {((0) + 1)}^{3}}$

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 $0$ .

$f'' (0) = \frac{2 (3 \cdot 0 - 18 \cdot 0 + 43)}{{(0 - 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = \frac{2 (0 - 18 \cdot 0 + 43)}{{(0 - 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.1.3

Multiply $- 18$ by $0$ .

$f'' (0) = \frac{2 (0 + 0 + 43)}{{(0 - 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.1.4

Add $0$ and $0$ .

$f'' (0) = \frac{2 (0 + 43)}{{(0 - 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.1.5

Add $0$ and $43$ .

$f'' (0) = \frac{2 \cdot 43}{{(0 - 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Subtract $7$ from $0$ .

$f'' (0) = \frac{2 \cdot 43}{{(- 7)}^{3} {(0 + 1)}^{3}}$

Step 5.2.2.2

Add $0$ and $1$ .

$f'' (0) = \frac{2 \cdot 43}{{(- 7)}^{3} \cdot 1^{3}}$

Step 5.2.2.3

Raise $- 7$ to the power of $3$ .

$f'' (0) = \frac{2 \cdot 43}{- 343 \cdot 1^{3}}$

Step 5.2.2.4

One to any power is one.

$f'' (0) = \frac{2 \cdot 43}{- 343 \cdot 1}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $43$ .

$f'' (0) = \frac{86}{- 343 \cdot 1}$

Step 5.2.3.2

Multiply $- 343$ by $1$ .

$f'' (0) = \frac{86}{- 343}$

Step 5.2.3.3

Move the negative in front of the fraction.

$f'' (0) = - \frac{86}{343}$

Step 5.2.4

The final answer is $- \frac{86}{343}$ .

$- \frac{86}{343}$

Step 5.3

The graph is concave down on the interval $(- 1, 7)$ because $f'' (0)$ is negative.

Concave down on $(- 1, 7)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(7, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $10$ in the expression.

$f'' (10) = \frac{2 (3 {(10)}^{2} - 18 \cdot 10 + 43)}{{((10) - 7)}^{3} {((10) + 1)}^{3}}$

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

Multiply $- 18$ by $10$ .

$f'' (10) = \frac{2 (3 \cdot 10^{2} - 180 + 43)}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.2

Add $- 180$ and $43$ .

$f'' (10) = \frac{2 (3 \cdot 10^{2} - 137)}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.3

Convert $- 137$ to scientific notation.

$f'' (10) = \frac{2 (3 \cdot 10^{2} - 1.37 \cdot 10^{2})}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.4

Factor $10^{2}$ out of $3 \cdot 10^{2} - 1.37 \cdot 10^{2}$ .

$f'' (10) = \frac{2 ((3 - 1.37) \cdot 10^{2})}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.5

Subtract $1.37$ from $3$ .

$f'' (10) = \frac{2 \cdot 1.63 \cdot 10^{2}}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.6

Multiply $2$ by $1.63$ .

$f'' (10) = \frac{3.26 \cdot 10^{2}}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.1.7

Raise $10$ to the power of $2$ .

$f'' (10) = \frac{3.26 \cdot 100}{{(10 - 7)}^{3} {(10 + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Subtract $7$ from $10$ .

$f'' (10) = \frac{3.26 \cdot 100}{3^{3} {(10 + 1)}^{3}}$

Step 6.2.2.2

Add $10$ and $1$ .

$f'' (10) = \frac{3.26 \cdot 100}{3^{3} \cdot 11^{3}}$

Step 6.2.2.3

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

$f'' (10) = \frac{3.26 \cdot 100}{27 \cdot 11^{3}}$

Step 6.2.2.4

Raise $11$ to the power of $3$ .

$f'' (10) = \frac{3.26 \cdot 100}{27 \cdot 1331}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $3.26$ by $100$ .

$f'' (10) = \frac{326}{27 \cdot 1331}$

Step 6.2.3.2

Multiply $27$ by $1331$ .

$f'' (10) = \frac{326}{35937}$

Step 6.2.3.3

Cancel the common factor of $326$ and $35937$ .

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Step 6.2.3.3.1

Rewrite $326$ as $1 (326)$ .

$f'' (10) = \frac{1 (326)}{35937}$

Step 6.2.3.3.2

Cancel the common factors.

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Step 6.2.3.3.2.1

Rewrite $35937$ as $1 (35937)$ .

$f'' (10) = \frac{1 \cdot 326}{1 \cdot 35937}$

Step 6.2.3.3.2.2

Cancel the common factor.

$f'' (10) = \frac{1 \cdot 326}{1 \cdot 35937}$

Step 6.2.3.3.2.3

Rewrite the expression.

$f'' (10) = \frac{326}{35937}$

Step 6.2.4

The final answer is $\frac{326}{35937}$ .

$\frac{326}{35937}$

Step 6.3

The graph is concave up on the interval $(7, \infty)$ because $f'' (10)$ is positive.

Concave up on $(7, \infty)$ 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 $(- \infty, - 1)$ since $f'' (x)$ is positive

Concave down on $(- 1, 7)$ since $f'' (x)$ is negative

Concave up on $(7, \infty)$ since $f'' (x)$ is positive

Step 8