Calculus Examples

$f (x) = \ln (x^{2} - 8 x + 41)$

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} - 8 x + 41$ .

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Step 1.1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 8 x + 41$ .

$f' (x) = \frac{d}{d u} (\ln (u)) \frac{d}{d x} (x^{2} - 8 x + 41)$

Step 1.1.1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$f' (x) = \frac{1}{u} \cdot \frac{d}{d x} (x^{2} - 8 x + 41)$

Step 1.1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 8 x + 41$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot \frac{d}{d x} (x^{2} - 8 x + 41)$

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 8 x + 41$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41]$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (41))$

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 $n = 2$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (41))$

Step 1.1.1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x - 8 \frac{d}{d x} x + \frac{d}{d x} (41))$

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 $n = 1$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x - 8 \cdot 1 + \frac{d}{d x} (41))$

Step 1.1.1.2.5

Multiply $- 8$ by $1$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x - 8 + \frac{d}{d x} (41))$

Step 1.1.1.2.6

Since $41$ is constant with respect to $x$ , the derivative of $41$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x - 8 + 0)$

Step 1.1.1.2.7

Add $2 x - 8$ and $0$ .

$f' (x) = \frac{1}{x^{2} - 8 x + 41} \cdot (2 x - 8)$

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Reorder the factors of $\frac{1}{x^{2} - 8 x + 41} (2 x - 8)$ .

$f' (x) = (2 x - 8) (\frac{1}{x^{2} - 8 x + 41})$

Step 1.1.1.3.2

Multiply $2 x - 8$ by $\frac{1}{x^{2} - 8 x + 41}$ .

$f' (x) = \frac{2 x - 8}{x^{2} - 8 x + 41}$

Step 1.1.1.3.3

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

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Step 1.1.1.3.3.1

Factor $2$ out of $2 x$ .

$f' (x) = \frac{2 (x) - 8}{x^{2} - 8 x + 41}$

Step 1.1.1.3.3.2

Factor $2$ out of $- 8$ .

$f' (x) = \frac{2 x + 2 \cdot - 4}{x^{2} - 8 x + 41}$

Step 1.1.1.3.3.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

$f' (x) = \frac{2 (x - 4)}{x^{2} - 8 x + 41}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 (x - 4)}{x^{2} - 8 x + 41}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x - 4}{x^{2} - 8 x + 41}]$ .

$f'' (x) = 2 \frac{d}{d x} (\frac{x - 4}{x^{2} - 8 x + 41})$

Step 1.1.2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 4$ and $g (x) = x^{2} - 8 x + 41$ .

$f'' (x) = 2 (\frac{(x^{2} - 8 x + 41) \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

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

$f'' (x) = 2 (\frac{(x^{2} - 8 x + 41) (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.2

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

$f'' (x) = 2 (\frac{(x^{2} - 8 x + 41) (1 + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.3

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

$f'' (x) = 2 (\frac{(x^{2} - 8 x + 41) (1 + 0) - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.4

Simplify the expression.

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Step 1.1.2.3.4.1

Add $1$ and $0$ .

$f'' (x) = 2 (\frac{(x^{2} - 8 x + 41) \cdot 1 - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.4.2

Multiply $x^{2} - 8 x + 41$ by $1$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) \frac{d}{d x} (x^{2} - 8 x + 41)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.5

By the Sum Rule, the derivative of $x^{2} - 8 x + 41$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41]$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (41))}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.6

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

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (41))}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.7

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 \frac{d}{d x} x + \frac{d}{d x} (41))}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.8

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

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 \cdot 1 + \frac{d}{d x} (41))}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.9

Multiply $- 8$ by $1$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 + \frac{d}{d x} (41))}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.10

Since $41$ is constant with respect to $x$ , the derivative of $41$ with respect to $x$ is $0$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 + 0)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.11

Combine fractions.

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Step 1.1.2.3.11.1

Add $2 x - 8$ and $0$ .

$f'' (x) = 2 (\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8)}{{(x^{2} - 8 x + 41)}^{2}})$

Step 1.1.2.3.11.2

Combine $2$ and $\frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8)}{{(x^{2} - 8 x + 41)}^{2}}$ .

$f'' (x) = \frac{2 (x^{2} - 8 x + 41 - (x - 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Apply the distributive property.

$f'' (x) = \frac{2 (x^{2} - 8 x + 41 + (- x + 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.2

Apply the distributive property.

$f'' (x) = \frac{2 x^{2} + 2 (- 8 x) + 2 \cdot 41 + 2 ((- x + 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3

Simplify the numerator.

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Step 1.1.2.4.3.1

Simplify each term.

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Step 1.1.2.4.3.1.1

Multiply $- 8$ by $2$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 2 \cdot 41 + 2 ((- x + 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.2

Multiply $2$ by $41$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 ((- x + 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.3

Multiply $- 1$ by $- 4$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 ((- x + 4) (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.4

Expand $(- x + 4) (2 x - 8)$ using the FOIL Method.

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Step 1.1.2.4.3.1.4.1

Apply the distributive property.

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- x (2 x - 8) + 4 (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.4.2

Apply the distributive property.

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- x (2 x) - x \cdot - 8 + 4 (2 x - 8))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.4.3

Apply the distributive property.

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- x (2 x) - x \cdot - 8 + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5

Simplify and combine like terms.

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Step 1.1.2.4.3.1.5.1

Simplify each term.

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Step 1.1.2.4.3.1.5.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 1 \cdot (2 x \cdot x) - x \cdot - 8 + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.2

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

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Step 1.1.2.4.3.1.5.1.2.1

Move $x$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 1 \cdot (2 (x \cdot x)) - x \cdot - 8 + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 1 \cdot (2 x^{2}) - x \cdot - 8 + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.3

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2} - x \cdot - 8 + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.4

Multiply $- 8$ by $- 1$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2} + 8 x + 4 (2 x) + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.5

Multiply $2$ by $4$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2} + 8 x + 8 x + 4 \cdot - 8)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.1.6

Multiply $4$ by $- 8$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2} + 8 x + 8 x - 32)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.5.2

Add $8 x$ and $8 x$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2} + 16 x - 32)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.6

Apply the distributive property.

$f'' (x) = \frac{2 x^{2} - 16 x + 82 + 2 (- 2 x^{2}) + 2 (16 x) + 2 \cdot - 32}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.7

Simplify.

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Step 1.1.2.4.3.1.7.1

Multiply $- 2$ by $2$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 - 4 x^{2} + 2 (16 x) + 2 \cdot - 32}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.7.2

Multiply $16$ by $2$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 - 4 x^{2} + 32 x + 2 \cdot - 32}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.1.7.3

Multiply $2$ by $- 32$ .

$f'' (x) = \frac{2 x^{2} - 16 x + 82 - 4 x^{2} + 32 x - 64}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.2

Subtract $4 x^{2}$ from $2 x^{2}$ .

$f'' (x) = \frac{- 2 x^{2} - 16 x + 82 + 32 x - 64}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.3

Add $- 16 x$ and $32 x$ .

$f'' (x) = \frac{- 2 x^{2} + 16 x + 82 - 64}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.3.4

Subtract $64$ from $82$ .

$f'' (x) = \frac{- 2 x^{2} + 16 x + 18}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4

Simplify the numerator.

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Step 1.1.2.4.4.1

Factor $2$ out of $- 2 x^{2} + 16 x + 18$ .

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Step 1.1.2.4.4.1.1

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

$f'' (x) = \frac{2 (- x^{2}) + 16 x + 18}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.1.2

Factor $2$ out of $16 x$ .

$f'' (x) = \frac{2 (- x^{2}) + 2 (8 x) + 18}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.1.3

Factor $2$ out of $18$ .

$f'' (x) = \frac{2 (- x^{2}) + 2 (8 x) + 2 (9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.1.4

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

$f'' (x) = \frac{2 (- x^{2} + 8 x) + 2 (9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.1.5

Factor $2$ out of $2 (- x^{2} + 8 x) + 2 (9)$ .

$f'' (x) = \frac{2 (- x^{2} + 8 x + 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2

Factor by grouping.

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Step 1.1.2.4.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 9 = - 9$ and whose sum is $b = 8$ .

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Step 1.1.2.4.4.2.1.1

Factor $8$ out of $8 x$ .

$f'' (x) = \frac{2 (- x^{2} + 8 (x) + 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2.1.2

Rewrite $8$ as $- 1$ plus $9$

$f'' (x) = \frac{2 (- x^{2} + (- 1 + 9) x + 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2.1.3

Apply the distributive property.

$f'' (x) = \frac{2 (- x^{2} - 1 x + 9 x + 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2.2

Factor out the greatest common factor from each group.

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Step 1.1.2.4.4.2.2.1

Group the first two terms and the last two terms.

$f'' (x) = \frac{2 ((- x^{2} - 1 x) + 9 x + 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2.2.2

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

$f'' (x) = \frac{2 (x (- x - 1) - 9 (- x - 1))}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$f'' (x) = \frac{2 ((- x - 1) (x - 9))}{{(x^{2} - 8 x + 41)}^{2}}$

$f'' (x) = \frac{2 (- x - 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.5

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

$f'' (x) = \frac{2 (- (x) - 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.6

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

$f'' (x) = \frac{2 (- (x) - 1 \cdot 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.7

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

$f'' (x) = \frac{2 (- (x + 1)) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.8

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$f'' (x) = \frac{2 (- 1 (x + 1)) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{(2 (x + 1)) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.2.4.10

Reorder factors in $- \frac{(2 (x + 1)) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$ .

$f'' (x) = - \frac{2 (x + 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2 (x + 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$ .

$- \frac{2 (x + 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2 (x + 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{2 (x + 1) (x - 9)}{{(x^{2} - 8 x + 41)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 (x + 1) (x - 9) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

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

$x + 1 = 0$

$x - 9 = 0$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x + 1 = 0$

Step 1.2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.3.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 1.2.3.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 1.2.3.4

The final solution is all the values that make $2 (x + 1) (x - 9) = 0$ true.

$x = - 1, 9$

Step 2

Find the domain of $f (x) = \ln (x^{2} - 8 x + 41)$ .

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Step 2.1

Set the argument in $\ln (x^{2} - 8 x + 41)$ greater than $0$ to find where the expression is defined.

$x^{2} - 8 x + 41 > 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Convert the inequality to an equation.

$x^{2} - 8 x + 41 = 0$

Step 2.2.2

Use the quadratic formula to find the solutions.

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

Step 2.2.3

Substitute the values $a = 1$ , $b = - 8$ , and $c = 41$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 41)}}{2 \cdot 1}$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 2.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 2.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 41}}{2 \cdot 1}$

Step 2.2.4.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{8 \pm \sqrt{64 - 164}}{2 \cdot 1}$

Step 2.2.4.1.3

Subtract $164$ from $64$ .

$x = \frac{8 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 2.2.4.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 2.2.4.1.5

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.4.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.4.1.7

Rewrite $100$ as $10^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 2.2.4.1.8

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

$x = \frac{8 \pm i \cdot 10}{2 \cdot 1}$

Step 2.2.4.1.9

Move $10$ to the left of $i$ .

$x = \frac{8 \pm 10 i}{2 \cdot 1}$

Step 2.2.4.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 10 i}{2}$

Step 2.2.4.3

Simplify $\frac{8 \pm 10 i}{2}$ .

$x = 4 \pm 5 i$

Step 2.2.5

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

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 2.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 2.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 41}}{2 \cdot 1}$

Step 2.2.5.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{8 \pm \sqrt{64 - 164}}{2 \cdot 1}$

Step 2.2.5.1.3

Subtract $164$ from $64$ .

$x = \frac{8 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 2.2.5.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 2.2.5.1.5

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.5.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.5.1.7

Rewrite $100$ as $10^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 2.2.5.1.8

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

$x = \frac{8 \pm i \cdot 10}{2 \cdot 1}$

Step 2.2.5.1.9

Move $10$ to the left of $i$ .

$x = \frac{8 \pm 10 i}{2 \cdot 1}$

Step 2.2.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 10 i}{2}$

Step 2.2.5.3

Simplify $\frac{8 \pm 10 i}{2}$ .

$x = 4 \pm 5 i$

Step 2.2.5.4

Change the $\pm$ to $+$ .

$x = 4 + 5 i$

Step 2.2.6

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

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 41}}{2 \cdot 1}$

Step 2.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 41$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 41}}{2 \cdot 1}$

Step 2.2.6.1.2.2

Multiply $- 4$ by $41$ .

$x = \frac{8 \pm \sqrt{64 - 164}}{2 \cdot 1}$

Step 2.2.6.1.3

Subtract $164$ from $64$ .

$x = \frac{8 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 2.2.6.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 2.2.6.1.5

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.6.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 2.2.6.1.7

Rewrite $100$ as $10^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 2.2.6.1.8

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

$x = \frac{8 \pm i \cdot 10}{2 \cdot 1}$

Step 2.2.6.1.9

Move $10$ to the left of $i$ .

$x = \frac{8 \pm 10 i}{2 \cdot 1}$

Step 2.2.6.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 10 i}{2}$

Step 2.2.6.3

Simplify $\frac{8 \pm 10 i}{2}$ .

$x = 4 \pm 5 i$

Step 2.2.6.4

Change the $\pm$ to $-$ .

$x = 4 - 5 i$

Step 2.2.7

Identify the leading coefficient.

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Step 2.2.7.1

The leading term in a polynomial is the term with the highest degree.

$x^{2}$

Step 2.2.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 2.2.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} - 8 x + 41$ is always greater than $0$ .

All real numbers

Step 2.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

$(- \infty, - 1) \cup (- 1, 9) \cup (9, \infty)$

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.

$f'' (- 4) = - \frac{2 ((- 4) + 1) ((- 4) - 9)}{{({(- 4)}^{2} - 8 \cdot - 4 + 41)}^{2}}$

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

Add $- 4$ and $1$ .

$f'' (- 4) = - \frac{2 \cdot (- 3 (- 4 - 9))}{{({(- 4)}^{2} - 8 \cdot - 4 + 41)}^{2}}$

Step 4.2.1.2

Multiply $2$ by $- 3$ .

$f'' (- 4) = - \frac{- 6 (- 4 - 9)}{{({(- 4)}^{2} - 8 \cdot - 4 + 41)}^{2}}$

Step 4.2.1.3

Subtract $9$ from $- 4$ .

$f'' (- 4) = - \frac{- 6 \cdot - 13}{{({(- 4)}^{2} - 8 \cdot - 4 + 41)}^{2}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$f'' (- 4) = - \frac{- 6 \cdot - 13}{{(16 - 8 \cdot - 4 + 41)}^{2}}$

Step 4.2.2.2

Multiply $- 8$ by $- 4$ .

$f'' (- 4) = - \frac{- 6 \cdot - 13}{{(16 + 32 + 41)}^{2}}$

Step 4.2.2.3

Add $16$ and $32$ .

$f'' (- 4) = - \frac{- 6 \cdot - 13}{{(48 + 41)}^{2}}$

Step 4.2.2.4

Add $48$ and $41$ .

$f'' (- 4) = - \frac{- 6 \cdot - 13}{89^{2}}$

Step 4.2.2.5

Raise $89$ to the power of $2$ .

$f'' (- 4) = - \frac{- 6 \cdot - 13}{7921}$

Step 4.2.3

Multiply $- 6$ by $- 13$ .

$f'' (- 4) = - \frac{78}{7921}$

Step 4.2.4

The final answer is $- \frac{78}{7921}$ .

$- \frac{78}{7921}$

Step 4.3

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

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

Step 5

Substitute any number from the interval $(- 1, 9)$ 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.

$f'' (0) = - \frac{2 ((0) + 1) ((0) - 9)}{{({(0)}^{2} - 8 \cdot 0 + 41)}^{2}}$

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

Add $0$ and $1$ .

$f'' (0) = - \frac{2 \cdot (1 (0 - 9))}{{(0^{2} - 8 \cdot 0 + 41)}^{2}}$

Step 5.2.1.2

Multiply $2$ by $1$ .

$f'' (0) = - \frac{2 (0 - 9)}{{(0^{2} - 8 \cdot 0 + 41)}^{2}}$

Step 5.2.1.3

Subtract $9$ from $0$ .

$f'' (0) = - \frac{2 \cdot - 9}{{(0^{2} - 8 \cdot 0 + 41)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

$f'' (0) = - \frac{2 \cdot - 9}{{(0 - 8 \cdot 0 + 41)}^{2}}$

Step 5.2.2.2

Multiply $- 8$ by $0$ .

$f'' (0) = - \frac{2 \cdot - 9}{{(0 + 0 + 41)}^{2}}$

Step 5.2.2.3

Add $0$ and $0$ .

$f'' (0) = - \frac{2 \cdot - 9}{{(0 + 41)}^{2}}$

Step 5.2.2.4

Add $0$ and $41$ .

$f'' (0) = - \frac{2 \cdot - 9}{41^{2}}$

Step 5.2.2.5

Raise $41$ to the power of $2$ .

$f'' (0) = - \frac{2 \cdot - 9}{1681}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $- 9$ .

$f'' (0) = - \frac{- 18}{1681}$

Step 5.2.3.2

Move the negative in front of the fraction.

$f'' (0) = \frac{18}{1681}$

Step 5.2.4

The final answer is $\frac{18}{1681}$ .

$\frac{18}{1681}$

Step 5.3

The graph is concave up on the interval $(- 1, 9)$ because $f'' (0)$ is positive.

Concave up on $(- 1, 9)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(9, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $12$ in the expression.

$f'' (12) = - \frac{2 ((12) + 1) ((12) - 9)}{{({(12)}^{2} - 8 \cdot 12 + 41)}^{2}}$

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

Add $12$ and $1$ .

$f'' (12) = - \frac{2 \cdot (13 (12 - 9))}{{(12^{2} - 8 \cdot 12 + 41)}^{2}}$

Step 6.2.1.2

Multiply $2$ by $13$ .

$f'' (12) = - \frac{26 (12 - 9)}{{(12^{2} - 8 \cdot 12 + 41)}^{2}}$

Step 6.2.1.3

Subtract $9$ from $12$ .

$f'' (12) = - \frac{26 \cdot 3}{{(12^{2} - 8 \cdot 12 + 41)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (12) = - \frac{26 \cdot 3}{{(144 - 8 \cdot 12 + 41)}^{2}}$

Step 6.2.2.2

Multiply $- 8$ by $12$ .

$f'' (12) = - \frac{26 \cdot 3}{{(144 - 96 + 41)}^{2}}$

Step 6.2.2.3

Subtract $96$ from $144$ .

$f'' (12) = - \frac{26 \cdot 3}{{(48 + 41)}^{2}}$

Step 6.2.2.4

Add $48$ and $41$ .

$f'' (12) = - \frac{26 \cdot 3}{89^{2}}$

Step 6.2.2.5

Raise $89$ to the power of $2$ .

$f'' (12) = - \frac{26 \cdot 3}{7921}$

Step 6.2.3

Multiply $26$ by $3$ .

$f'' (12) = - \frac{78}{7921}$

Step 6.2.4

The final answer is $- \frac{78}{7921}$ .

$- \frac{78}{7921}$

Step 6.3

The graph is concave down on the interval $(9, \infty)$ because $f'' (12)$ is negative.

Concave down on $(9, \infty)$ 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 down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Concave up on $(- 1, 9)$ since $f'' (x)$ is positive

Concave down on $(9, \infty)$ since $f'' (x)$ is negative

Step 8