Calculus Examples

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

Step 1

Write $\ln (x^{2} - 8 x + 41)$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 2.1.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} - 8 x + 41]$

Step 2.1.1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2} - 8 x + 41]$

Step 2.1.1.3

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

$\frac{1}{x^{2} - 8 x + 41} \frac{d}{d x} [x^{2} - 8 x + 41]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$\frac{1}{x^{2} - 8 x + 41} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41])$

Step 2.1.2.2

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

$\frac{1}{x^{2} - 8 x + 41} (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41])$

Step 2.1.2.3

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

$\frac{1}{x^{2} - 8 x + 41} (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [41])$

Step 2.1.2.4

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

$\frac{1}{x^{2} - 8 x + 41} (2 x - 8 \cdot 1 + \frac{d}{d x} [41])$

Step 2.1.2.5

Multiply $- 8$ by $1$ .

$\frac{1}{x^{2} - 8 x + 41} (2 x - 8 + \frac{d}{d x} [41])$

Step 2.1.2.6

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

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

Step 2.1.2.7

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

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

Step 2.1.3

Simplify.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

$\frac{2 x - 8}{x^{2} - 8 x + 41}$

Step 2.1.3.3

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

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Step 2.1.3.3.1

Factor $2$ out of $2 x$ .

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

Step 2.1.3.3.2

Factor $2$ out of $- 8$ .

$\frac{2 x + 2 \cdot - 4}{x^{2} - 8 x + 41}$

Step 2.1.3.3.3

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

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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 $2 \frac{d}{d x} [\frac{x - 4}{x^{2} - 8 x + 41}]$ .

$2 \frac{d}{d x} [\frac{x - 4}{x^{2} - 8 x + 41}]$

Step 2.2.2

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 $g (x) = x^{2} - 8 x + 41$ .

$2 \frac{(x^{2} - 8 x + 41) \frac{d}{d x} [x - 4] - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3

Differentiate.

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Step 2.2.3.1

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

$2 \frac{(x^{2} - 8 x + 41) (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.2

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

$2 \frac{(x^{2} - 8 x + 41) (1 + \frac{d}{d x} [- 4]) - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.3

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

$2 \frac{(x^{2} - 8 x + 41) (1 + 0) - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.4

Simplify the expression.

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Step 2.2.3.4.1

Add $1$ and $0$ .

$2 \frac{(x^{2} - 8 x + 41) \cdot 1 - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.4.2

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

$2 \frac{x^{2} - 8 x + 41 - (x - 4) \frac{d}{d x} [x^{2} - 8 x + 41]}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.5

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

$2 \frac{x^{2} - 8 x + 41 - (x - 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41])}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.6

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

$2 \frac{x^{2} - 8 x + 41 - (x - 4) (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [41])}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.7

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

$2 \frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [41])}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.8

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

$2 \frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 \cdot 1 + \frac{d}{d x} [41])}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.9

Multiply $- 8$ by $1$ .

$2 \frac{x^{2} - 8 x + 41 - (x - 4) (2 x - 8 + \frac{d}{d x} [41])}{{(x^{2} - 8 x + 41)}^{2}}$

Step 2.2.3.10

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

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

Step 2.2.3.11

Combine fractions.

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Step 2.2.3.11.1

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

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

Step 2.2.3.11.2

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

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

Step 2.2.4

Simplify.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Apply the distributive property.

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

Step 2.2.4.3

Simplify the numerator.

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Step 2.2.4.3.1

Simplify each term.

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Step 2.2.4.3.1.1

Multiply $- 8$ by $2$ .

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

Step 2.2.4.3.1.2

Multiply $2$ by $41$ .

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

Step 2.2.4.3.1.3

Multiply $- 1$ by $- 4$ .

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

Step 2.2.4.3.1.4

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

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Step 2.2.4.3.1.4.1

Apply the distributive property.

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

Step 2.2.4.3.1.4.2

Apply the distributive property.

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

Step 2.2.4.3.1.4.3

Apply the distributive property.

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

Step 2.2.4.3.1.5

Simplify and combine like terms.

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Step 2.2.4.3.1.5.1

Simplify each term.

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Step 2.2.4.3.1.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.4.3.1.5.1.2

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

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Step 2.2.4.3.1.5.1.2.1

Move $x$ .

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

Step 2.2.4.3.1.5.1.2.2

Multiply $x$ by $x$ .

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

Step 2.2.4.3.1.5.1.3

Multiply $- 1$ by $2$ .

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

Step 2.2.4.3.1.5.1.4

Multiply $- 8$ by $- 1$ .

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

Step 2.2.4.3.1.5.1.5

Multiply $2$ by $4$ .

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

Step 2.2.4.3.1.5.1.6

Multiply $4$ by $- 8$ .

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

Step 2.2.4.3.1.5.2

Add $8 x$ and $8 x$ .

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

Step 2.2.4.3.1.6

Apply the distributive property.

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

Step 2.2.4.3.1.7

Simplify.

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Step 2.2.4.3.1.7.1

Multiply $- 2$ by $2$ .

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

Step 2.2.4.3.1.7.2

Multiply $16$ by $2$ .

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

Step 2.2.4.3.1.7.3

Multiply $2$ by $- 32$ .

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

Step 2.2.4.3.2

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

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

Step 2.2.4.3.3

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

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

Step 2.2.4.3.4

Subtract $64$ from $82$ .

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

Step 2.2.4.4

Simplify the numerator.

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Step 2.2.4.4.1

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

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Step 2.2.4.4.1.1

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

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

Step 2.2.4.4.1.2

Factor $2$ out of $16 x$ .

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

Step 2.2.4.4.1.3

Factor $2$ out of $18$ .

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

Step 2.2.4.4.1.4

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

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

Step 2.2.4.4.1.5

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

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

Step 2.2.4.4.2

Factor by grouping.

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Step 2.2.4.4.2.1

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 2.2.4.4.2.1.1

Factor $8$ out of $8 x$ .

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

Step 2.2.4.4.2.1.2

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

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

Step 2.2.4.4.2.1.3

Apply the distributive property.

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

Step 2.2.4.4.2.2

Factor out the greatest common factor from each group.

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Step 2.2.4.4.2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.4.4.2.2.2

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

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

Step 2.2.4.4.2.3

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

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

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

Step 2.2.4.5

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

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

Step 2.2.4.6

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

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

Step 2.2.4.7

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

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

Step 2.2.4.8

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

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

Step 2.2.4.9

Move the negative in front of the fraction.

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

Step 2.2.4.10

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 2.3

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 3

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 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

Set the numerator equal to zero.

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

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

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 3.3.2

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

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Step 3.3.2.1

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

$x + 1 = 0$

Step 3.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.3.3

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

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Step 3.3.3.1

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

$x - 9 = 0$

Step 3.3.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 3.3.4

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

$x = - 1, 9$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $- 1$ in $f (x) = \ln (x^{2} - 8 x + 41)$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (- 1) = \ln (1 - 8 \cdot - 1 + 41)$

Step 4.1.2.1.2

Multiply $- 8$ by $- 1$ .

$f (- 1) = \ln (1 + 8 + 41)$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $1$ and $8$ .

$f (- 1) = \ln (9 + 41)$

Step 4.1.2.2.2

Add $9$ and $41$ .

$f (- 1) = \ln (50)$

Step 4.1.2.3

The final answer is $\ln (50)$ .

$\ln (50)$

Step 4.2

The point found by substituting $- 1$ in $f (x) = \ln (x^{2} - 8 x + 41)$ is $(- 1, \ln (50))$ . This point can be an inflection point.

$(- 1, \ln (50))$

Step 4.3

Substitute $9$ in $f (x) = \ln (x^{2} - 8 x + 41)$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $9$ in the expression.

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

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $9$ to the power of $2$ .

$f (9) = \ln (81 - 8 \cdot 9 + 41)$

Step 4.3.2.1.2

Multiply $- 8$ by $9$ .

$f (9) = \ln (81 - 72 + 41)$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Subtract $72$ from $81$ .

$f (9) = \ln (9 + 41)$

Step 4.3.2.2.2

Add $9$ and $41$ .

$f (9) = \ln (50)$

Step 4.3.2.3

The final answer is $\ln (50)$ .

$\ln (50)$

Step 4.4

The point found by substituting $9$ in $f (x) = \ln (x^{2} - 8 x + 41)$ is $(9, \ln (50))$ . This point can be an inflection point.

$(9, \ln (50))$

Step 4.5

Determine the points that could be inflection points.

$(- 1, \ln (50)), (9, \ln (50))$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 6

Substitute a value from the interval $(- \infty, - 1)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.1$ in the expression.

$f'' (- 1.1) = - \frac{2 ((- 1.1) + 1) ((- 1.1) - 9)}{{({(- 1.1)}^{2} - 8 \cdot - 1.1 + 41)}^{2}}$

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 $- 1.1$ and $1$ .

$f'' (- 1.1) = - \frac{2 \cdot (- 0.1 (- 1.1 - 9))}{{({(- 1.1)}^{2} - 8 \cdot - 1.1 + 41)}^{2}}$

Step 6.2.1.2

Combine exponents.

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Step 6.2.1.2.1

Multiply $2$ by $- 0.1$ .

$f'' (- 1.1) = - \frac{- 0.2 (- 1.1 - 9)}{{({(- 1.1)}^{2} - 8 \cdot - 1.1 + 41)}^{2}}$

Step 6.2.1.2.2

Multiply $- 0.2$ by $- 1.1 - 9$ .

$f'' (- 1.1) = - \frac{2.02}{{({(- 1.1)}^{2} - 8 \cdot - 1.1 + 41)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 1.1$ to the power of $2$ .

$f'' (- 1.1) = - \frac{2.02}{{(1.21 - 8 \cdot - 1.1 + 41)}^{2}}$

Step 6.2.2.2

Multiply $- 8$ by $- 1.1$ .

$f'' (- 1.1) = - \frac{2.02}{{(1.21 + 8.8 + 41)}^{2}}$

Step 6.2.2.3

Add $1.21$ and $8.8$ .

$f'' (- 1.1) = - \frac{2.02}{{(10.01 + 41)}^{2}}$

Step 6.2.2.4

Add $10.01$ and $41$ .

$f'' (- 1.1) = - \frac{2.02}{{51.01}^{2}}$

Step 6.2.2.5

Raise $51.01$ to the power of $2$ .

$f'' (- 1.1) = - \frac{2.02}{2602.0201}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $2.02$ by $2602.0201$ .

$f'' (- 1.1) = - 1 \cdot 0.00077631$

Step 6.2.3.2

Multiply $- 1$ by $0.00077631$ .

$f'' (- 1.1) = - 0.00077631$

Step 6.2.4

The final answer is $- 0.00077631$ .

$- 0.00077631$

Step 6.3

At $- 1.1$ , the second derivative is $- 0.00077631$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 1)$

Decreasing on $(- \infty, - 1)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 1, 9)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Add $4$ and $1$ .

$f'' (4) = - \frac{2 \cdot (5 (4 - 9))}{{(4^{2} - 8 \cdot 4 + 41)}^{2}}$

Step 7.2.1.2

Multiply $2$ by $5$ .

$f'' (4) = - \frac{10 (4 - 9)}{{(4^{2} - 8 \cdot 4 + 41)}^{2}}$

Step 7.2.1.3

Subtract $9$ from $4$ .

$f'' (4) = - \frac{10 \cdot - 5}{{(4^{2} - 8 \cdot 4 + 41)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f'' (4) = - \frac{10 \cdot - 5}{{(16 - 8 \cdot 4 + 41)}^{2}}$

Step 7.2.2.2

Multiply $- 8$ by $4$ .

$f'' (4) = - \frac{10 \cdot - 5}{{(16 - 32 + 41)}^{2}}$

Step 7.2.2.3

Subtract $32$ from $16$ .

$f'' (4) = - \frac{10 \cdot - 5}{{(- 16 + 41)}^{2}}$

Step 7.2.2.4

Add $- 16$ and $41$ .

$f'' (4) = - \frac{10 \cdot - 5}{25^{2}}$

Step 7.2.2.5

Raise $25$ to the power of $2$ .

$f'' (4) = - \frac{10 \cdot - 5}{625}$

Step 7.2.3

Reduce the expression by cancelling the common factors.

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Step 7.2.3.1

Multiply $10$ by $- 5$ .

$f'' (4) = - \frac{- 50}{625}$

Step 7.2.3.2

Cancel the common factor of $- 50$ and $625$ .

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Step 7.2.3.2.1

Factor $25$ out of $- 50$ .

$f'' (4) = - \frac{25 (- 2)}{625}$

Step 7.2.3.2.2

Cancel the common factors.

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Step 7.2.3.2.2.1

Factor $25$ out of $625$ .

$f'' (4) = - \frac{25 \cdot - 2}{25 \cdot 25}$

Step 7.2.3.2.2.2

Cancel the common factor.

$f'' (4) = - \frac{25 \cdot - 2}{25 \cdot 25}$

Step 7.2.3.2.2.3

Rewrite the expression.

$f'' (4) = - \frac{- 2}{25}$

Step 7.2.3.3

Move the negative in front of the fraction.

$f'' (4) = \frac{2}{25}$

Step 7.2.4

The final answer is $\frac{2}{25}$ .

$\frac{2}{25}$

Step 7.3

At $4$ , the second derivative is $0.08$ . Since this is positive, the second derivative is increasing on the interval $(- 1, 9)$ .

Increasing on $(- 1, 9)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(9, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $9.1$ in the expression.

$f'' (9.1) = - \frac{2 ((9.1) + 1) ((9.1) - 9)}{{({(9.1)}^{2} - 8 \cdot 9.1 + 41)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Add $9.1$ and $1$ .

$f'' (9.1) = - \frac{2 \cdot (10.1 (9.1 - 9))}{{({9.1}^{2} - 8 \cdot 9.1 + 41)}^{2}}$

Step 8.2.1.2

Combine exponents.

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Step 8.2.1.2.1

Multiply $2$ by $10.1$ .

$f'' (9.1) = - \frac{20.2 (9.1 - 9)}{{({9.1}^{2} - 8 \cdot 9.1 + 41)}^{2}}$

Step 8.2.1.2.2

Multiply $20.2$ by $9.1 - 9$ .

$f'' (9.1) = - \frac{2.02}{{({9.1}^{2} - 8 \cdot 9.1 + 41)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $9.1$ to the power of $2$ .

$f'' (9.1) = - \frac{2.02}{{(82.81 - 8 \cdot 9.1 + 41)}^{2}}$

Step 8.2.2.2

Multiply $- 8$ by $9.1$ .

$f'' (9.1) = - \frac{2.02}{{(82.81 - 72.8 + 41)}^{2}}$

Step 8.2.2.3

Subtract $72.8$ from $82.81$ .

$f'' (9.1) = - \frac{2.02}{{(10.01 + 41)}^{2}}$

Step 8.2.2.4

Add $10.01$ and $41$ .

$f'' (9.1) = - \frac{2.02}{{51.01}^{2}}$

Step 8.2.2.5

Raise $51.01$ to the power of $2$ .

$f'' (9.1) = - \frac{2.02}{2602.0201}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Divide $2.02$ by $2602.0201$ .

$f'' (9.1) = - 1 \cdot 0.00077631$

Step 8.2.3.2

Multiply $- 1$ by $0.00077631$ .

$f'' (9.1) = - 0.00077631$

Step 8.2.4

The final answer is $- 0.00077631$ .

$- 0.00077631$

Step 8.3

At $9.1$ , the second derivative is $- 0.00077631$ . Since this is negative, the second derivative is decreasing on the interval $(9, \infty)$

Decreasing on $(9, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 1, \ln (50)), (9, \ln (50))$ .

$(- 1, \ln (50)), (9, \ln (50))$

Step 10