Calculus Examples

$\ln (x^{2} - 4 x + 20)$

Step 1

Write $\ln (x^{2} - 4 x + 20)$ as a function.

$f (x) = \ln (x^{2} - 4 x + 20)$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $g (x) = x^{2} - 4 x + 20$ .

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Step 2.1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 4 x + 20$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} - 4 x + 20]$

Step 2.1.1.1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2} - 4 x + 20]$

Step 2.1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 4 x + 20$ .

$\frac{1}{x^{2} - 4 x + 20} \frac{d}{d x} [x^{2} - 4 x + 20]$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 4 x + 20$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20]$ .

$\frac{1}{x^{2} - 4 x + 20} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20])$

Step 2.1.1.2.2

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

$\frac{1}{x^{2} - 4 x + 20} (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20])$

Step 2.1.1.2.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$\frac{1}{x^{2} - 4 x + 20} (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [20])$

Step 2.1.1.2.4

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

$\frac{1}{x^{2} - 4 x + 20} (2 x - 4 \cdot 1 + \frac{d}{d x} [20])$

Step 2.1.1.2.5

Multiply $- 4$ by $1$ .

$\frac{1}{x^{2} - 4 x + 20} (2 x - 4 + \frac{d}{d x} [20])$

Step 2.1.1.2.6

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$\frac{1}{x^{2} - 4 x + 20} (2 x - 4 + 0)$

Step 2.1.1.2.7

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

$\frac{1}{x^{2} - 4 x + 20} (2 x - 4)$

Step 2.1.1.3

Simplify.

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Step 2.1.1.3.1

Reorder the factors of $\frac{1}{x^{2} - 4 x + 20} (2 x - 4)$ .

$(2 x - 4) \frac{1}{x^{2} - 4 x + 20}$

Step 2.1.1.3.2

Multiply $2 x - 4$ by $\frac{1}{x^{2} - 4 x + 20}$ .

$\frac{2 x - 4}{x^{2} - 4 x + 20}$

Step 2.1.1.3.3

Factor $2$ out of $2 x - 4$ .

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Step 2.1.1.3.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{x^{2} - 4 x + 20}$

Step 2.1.1.3.3.2

Factor $2$ out of $- 4$ .

$\frac{2 x + 2 \cdot - 2}{x^{2} - 4 x + 20}$

Step 2.1.1.3.3.3

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

$f' (x) = \frac{2 (x - 2)}{x^{2} - 4 x + 20}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 (x - 2)}{x^{2} - 4 x + 20}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x - 2}{x^{2} - 4 x + 20}]$ .

$2 \frac{d}{d x} [\frac{x - 2}{x^{2} - 4 x + 20}]$

Step 2.1.2.2

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 $f (x) = x - 2$ and $g (x) = x^{2} - 4 x + 20$ .

$2 \frac{(x^{2} - 4 x + 20) \frac{d}{d x} [x - 2] - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

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

$2 \frac{(x^{2} - 4 x + 20) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.2

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

$2 \frac{(x^{2} - 4 x + 20) (1 + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.3

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

$2 \frac{(x^{2} - 4 x + 20) (1 + 0) - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.4

Simplify the expression.

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Step 2.1.2.3.4.1

Add $1$ and $0$ .

$2 \frac{(x^{2} - 4 x + 20) \cdot 1 - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.4.2

Multiply $x^{2} - 4 x + 20$ by $1$ .

$2 \frac{x^{2} - 4 x + 20 - (x - 2) \frac{d}{d x} [x^{2} - 4 x + 20]}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.5

By the Sum Rule, the derivative of $x^{2} - 4 x + 20$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20]$ .

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20])}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.6

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

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [20])}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.7

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [20])}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.8

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

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4 \cdot 1 + \frac{d}{d x} [20])}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.9

Multiply $- 4$ by $1$ .

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4 + \frac{d}{d x} [20])}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.10

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4 + 0)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.11

Combine fractions.

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Step 2.1.2.3.11.1

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

$2 \frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.3.11.2

Combine $2$ and $\frac{x^{2} - 4 x + 20 - (x - 2) (2 x - 4)}{{(x^{2} - 4 x + 20)}^{2}}$ .

$\frac{2 (x^{2} - 4 x + 20 - (x - 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4

Simplify.

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Step 2.1.2.4.1

Apply the distributive property.

$\frac{2 (x^{2} - 4 x + 20 + (- x - - 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.2

Apply the distributive property.

$\frac{2 x^{2} + 2 (- 4 x) + 2 \cdot 20 + 2 ((- x - - 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3

Simplify the numerator.

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Step 2.1.2.4.3.1

Simplify each term.

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Step 2.1.2.4.3.1.1

Multiply $- 4$ by $2$ .

$\frac{2 x^{2} - 8 x + 2 \cdot 20 + 2 ((- x - - 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.2

Multiply $2$ by $20$ .

$\frac{2 x^{2} - 8 x + 40 + 2 ((- x - - 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.3

Multiply $- 1$ by $- 2$ .

$\frac{2 x^{2} - 8 x + 40 + 2 ((- x + 2) (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.4

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

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Step 2.1.2.4.3.1.4.1

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 40 + 2 (- x (2 x - 4) + 2 (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.4.2

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 40 + 2 (- x (2 x) - x \cdot - 4 + 2 (2 x - 4))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.4.3

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 40 + 2 (- x (2 x) - x \cdot - 4 + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5

Simplify and combine like terms.

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Step 2.1.2.4.3.1.5.1

Simplify each term.

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Step 2.1.2.4.3.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - 8 x + 40 + 2 (- 1 \cdot 2 x \cdot x - x \cdot - 4 + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.2

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

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Step 2.1.2.4.3.1.5.1.2.1

Move $x$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 1 \cdot 2 (x \cdot x) - x \cdot - 4 + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 1 \cdot 2 x^{2} - x \cdot - 4 + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2} - x \cdot - 4 + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.4

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2} + 4 x + 2 (2 x) + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.5

Multiply $2$ by $2$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2} + 4 x + 4 x + 2 \cdot - 4)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.1.6

Multiply $2$ by $- 4$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2} + 4 x + 4 x - 8)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.5.2

Add $4 x$ and $4 x$ .

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2} + 8 x - 8)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.6

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 40 + 2 (- 2 x^{2}) + 2 (8 x) + 2 \cdot - 8}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.7

Simplify.

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Step 2.1.2.4.3.1.7.1

Multiply $- 2$ by $2$ .

$\frac{2 x^{2} - 8 x + 40 - 4 x^{2} + 2 (8 x) + 2 \cdot - 8}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.7.2

Multiply $8$ by $2$ .

$\frac{2 x^{2} - 8 x + 40 - 4 x^{2} + 16 x + 2 \cdot - 8}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.1.7.3

Multiply $2$ by $- 8$ .

$\frac{2 x^{2} - 8 x + 40 - 4 x^{2} + 16 x - 16}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.2

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

$\frac{- 2 x^{2} - 8 x + 40 + 16 x - 16}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.3

Add $- 8 x$ and $16 x$ .

$\frac{- 2 x^{2} + 8 x + 40 - 16}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.3.4

Subtract $16$ from $40$ .

$\frac{- 2 x^{2} + 8 x + 24}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4

Simplify the numerator.

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Step 2.1.2.4.4.1

Factor $2$ out of $- 2 x^{2} + 8 x + 24$ .

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Step 2.1.2.4.4.1.1

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

$\frac{2 (- x^{2}) + 8 x + 24}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.1.2

Factor $2$ out of $8 x$ .

$\frac{2 (- x^{2}) + 2 (4 x) + 24}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.1.3

Factor $2$ out of $24$ .

$\frac{2 (- x^{2}) + 2 (4 x) + 2 (12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.1.4

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

$\frac{2 (- x^{2} + 4 x) + 2 (12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.1.5

Factor $2$ out of $2 (- x^{2} + 4 x) + 2 (12)$ .

$\frac{2 (- x^{2} + 4 x + 12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2

Factor by grouping.

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Step 2.1.2.4.4.2.1

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = 4$ .

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Step 2.1.2.4.4.2.1.1

Factor $4$ out of $4 x$ .

$\frac{2 (- x^{2} + 4 (x) + 12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2.1.2

Rewrite $4$ as $- 2$ plus $6$

$\frac{2 (- x^{2} + (- 2 + 6) x + 12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2.1.3

Apply the distributive property.

$\frac{2 (- x^{2} - 2 x + 6 x + 12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2.2

Factor out the greatest common factor from each group.

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Step 2.1.2.4.4.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((- x^{2} - 2 x) + 6 x + 12)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2.2.2

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

$\frac{2 (x (- x - 2) - 6 (- x - 2))}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$\frac{2 ((- x - 2) (x - 6))}{{(x^{2} - 4 x + 20)}^{2}}$

$\frac{2 (- x - 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.5

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

$\frac{2 (- (x) - 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.6

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

$\frac{2 (- (x) - 1 (2)) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.7

Factor $- 1$ out of $- (x) - 1 (2)$ .

$\frac{2 (- (x + 2)) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.8

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

$\frac{2 (- 1 (x + 2)) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.9

Move the negative in front of the fraction.

$- \frac{(2 (x + 2)) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.2.4.10

Reorder factors in $- \frac{(2 (x + 2)) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$ .

$f'' (x) = - \frac{2 (x + 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2 (x + 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$ .

$- \frac{2 (x + 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2 (x + 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- \frac{2 (x + 2) (x - 6)}{{(x^{2} - 4 x + 20)}^{2}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$2 (x + 2) (x - 6) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$x + 2 = 0$

$x - 6 = 0$

Step 2.2.3.2

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.2.3.2.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.2.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.3.3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 2.2.3.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 2.2.3.4

The final solution is all the values that make $2 (x + 2) (x - 6) = 0$ true.

$x = - 2, 6$

Step 3

Find the domain of $f (x) = \ln (x^{2} - 4 x + 20)$ .

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Step 3.1

Set the argument in $\ln (x^{2} - 4 x + 20)$ greater than $0$ to find where the expression is defined.

$x^{2} - 4 x + 20 > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Convert the inequality to an equation.

$x^{2} - 4 x + 20 = 0$

Step 3.2.2

Use the quadratic formula to find the solutions.

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

Step 3.2.3

Substitute the values $a = 1$ , $b = - 4$ , and $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Step 3.2.4

Simplify.

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Step 3.2.4.1

Simplify the numerator.

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Step 3.2.4.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 3.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 3.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 20}}{2 \cdot 1}$

Step 3.2.4.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot 1}$

Step 3.2.4.1.3

Subtract $80$ from $16$ .

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

Step 3.2.4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

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

Step 3.2.4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

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

Step 3.2.4.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 3.2.4.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 3.2.4.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot 1}$

Step 3.2.4.1.9

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

$x = \frac{4 \pm 8 i}{2 \cdot 1}$

Step 3.2.4.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 8 i}{2}$

Step 3.2.4.3

Simplify $\frac{4 \pm 8 i}{2}$ .

$x = 2 \pm 4 i$

Step 3.2.5

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

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Step 3.2.5.1

Simplify the numerator.

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Step 3.2.5.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 3.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 3.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 20}}{2 \cdot 1}$

Step 3.2.5.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot 1}$

Step 3.2.5.1.3

Subtract $80$ from $16$ .

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

Step 3.2.5.1.4

Rewrite $- 64$ as $- 1 (64)$ .

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

Step 3.2.5.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

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

Step 3.2.5.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 3.2.5.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 3.2.5.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot 1}$

Step 3.2.5.1.9

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

$x = \frac{4 \pm 8 i}{2 \cdot 1}$

Step 3.2.5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 8 i}{2}$

Step 3.2.5.3

Simplify $\frac{4 \pm 8 i}{2}$ .

$x = 2 \pm 4 i$

Step 3.2.5.4

Change the $\pm$ to $+$ .

$x = 2 + 4 i$

Step 3.2.6

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

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Step 3.2.6.1

Simplify the numerator.

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Step 3.2.6.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 3.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 3.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 20}}{2 \cdot 1}$

Step 3.2.6.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot 1}$

Step 3.2.6.1.3

Subtract $80$ from $16$ .

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

Step 3.2.6.1.4

Rewrite $- 64$ as $- 1 (64)$ .

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

Step 3.2.6.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

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

Step 3.2.6.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 3.2.6.1.7

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

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 3.2.6.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot 1}$

Step 3.2.6.1.9

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

$x = \frac{4 \pm 8 i}{2 \cdot 1}$

Step 3.2.6.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 8 i}{2}$

Step 3.2.6.3

Simplify $\frac{4 \pm 8 i}{2}$ .

$x = 2 \pm 4 i$

Step 3.2.6.4

Change the $\pm$ to $-$ .

$x = 2 - 4 i$

Step 3.2.7

Identify the leading coefficient.

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Step 3.2.7.1

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

$x^{2}$

Step 3.2.7.2

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

$1$

Step 3.2.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} - 4 x + 20$ is always greater than $0$ .

All real numbers

Step 3.3

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 4

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

$(- \infty, - 2) \cup (- 2, 6) \cup (6, \infty)$

Step 5

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

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Step 5.1

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

$f'' (- 4) = - \frac{2 ((- 4) + 2) ((- 4) - 6)}{{({(- 4)}^{2} - 4 \cdot - 4 + 20)}^{2}}$

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 $- 4$ and $2$ .

$f'' (- 4) = - \frac{2 \cdot (- 2 (- 4 - 6))}{{({(- 4)}^{2} - 4 \cdot - 4 + 20)}^{2}}$

Step 5.2.1.2

Multiply $2$ by $- 2$ .

$f'' (- 4) = - \frac{- 4 (- 4 - 6)}{{({(- 4)}^{2} - 4 \cdot - 4 + 20)}^{2}}$

Step 5.2.1.3

Subtract $6$ from $- 4$ .

$f'' (- 4) = - \frac{- 4 \cdot - 10}{{({(- 4)}^{2} - 4 \cdot - 4 + 20)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

$f'' (- 4) = - \frac{- 4 \cdot - 10}{{(16 - 4 \cdot - 4 + 20)}^{2}}$

Step 5.2.2.2

Multiply $- 4$ by $- 4$ .

$f'' (- 4) = - \frac{- 4 \cdot - 10}{{(16 + 16 + 20)}^{2}}$

Step 5.2.2.3

Add $16$ and $16$ .

$f'' (- 4) = - \frac{- 4 \cdot - 10}{{(32 + 20)}^{2}}$

Step 5.2.2.4

Add $32$ and $20$ .

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

Step 5.2.2.5

Raise $52$ to the power of $2$ .

$f'' (- 4) = - \frac{- 4 \cdot - 10}{2704}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Multiply $- 4$ by $- 10$ .

$f'' (- 4) = - \frac{40}{2704}$

Step 5.2.3.2

Cancel the common factor of $40$ and $2704$ .

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Step 5.2.3.2.1

Factor $8$ out of $40$ .

$f'' (- 4) = - \frac{8 (5)}{2704}$

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

Factor $8$ out of $2704$ .

$f'' (- 4) = - \frac{8 \cdot 5}{8 \cdot 338}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f'' (- 4) = - \frac{8 \cdot 5}{8 \cdot 338}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f'' (- 4) = - \frac{5}{338}$

Step 5.2.4

The final answer is $- \frac{5}{338}$ .

$- \frac{5}{338}$

Step 5.3

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

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

Step 6

Substitute any number from the interval $(- 2, 6)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (0) = - \frac{2 ((0) + 2) ((0) - 6)}{{({(0)}^{2} - 4 \cdot 0 + 20)}^{2}}$

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 $0$ and $2$ .

$f'' (0) = - \frac{2 \cdot (2 (0 - 6))}{{(0^{2} - 4 \cdot 0 + 20)}^{2}}$

Step 6.2.1.2

Multiply $2$ by $2$ .

$f'' (0) = - \frac{4 (0 - 6)}{{(0^{2} - 4 \cdot 0 + 20)}^{2}}$

Step 6.2.1.3

Subtract $6$ from $0$ .

$f'' (0) = - \frac{4 \cdot - 6}{{(0^{2} - 4 \cdot 0 + 20)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (0) = - \frac{4 \cdot - 6}{{(0 - 4 \cdot 0 + 20)}^{2}}$

Step 6.2.2.2

Multiply $- 4$ by $0$ .

$f'' (0) = - \frac{4 \cdot - 6}{{(0 + 0 + 20)}^{2}}$

Step 6.2.2.3

Add $0$ and $0$ .

$f'' (0) = - \frac{4 \cdot - 6}{{(0 + 20)}^{2}}$

Step 6.2.2.4

Add $0$ and $20$ .

$f'' (0) = - \frac{4 \cdot - 6}{20^{2}}$

Step 6.2.2.5

Raise $20$ to the power of $2$ .

$f'' (0) = - \frac{4 \cdot - 6}{400}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $4$ by $- 6$ .

$f'' (0) = - \frac{- 24}{400}$

Step 6.2.3.2

Cancel the common factor of $- 24$ and $400$ .

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Step 6.2.3.2.1

Factor $8$ out of $- 24$ .

$f'' (0) = - \frac{8 (- 3)}{400}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $8$ out of $400$ .

$f'' (0) = - \frac{8 \cdot - 3}{8 \cdot 50}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (0) = - \frac{8 \cdot - 3}{8 \cdot 50}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (0) = - \frac{- 3}{50}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f'' (0) = \frac{3}{50}$

Step 6.2.4

The final answer is $\frac{3}{50}$ .

$\frac{3}{50}$

Step 6.3

The graph is concave up on the interval $(- 2, 6)$ because $f'' (0)$ is positive.

Concave up on $(- 2, 6)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(6, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $8$ in the expression.

$f'' (8) = - \frac{2 ((8) + 2) ((8) - 6)}{{({(8)}^{2} - 4 \cdot 8 + 20)}^{2}}$

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 $8$ and $2$ .

$f'' (8) = - \frac{2 \cdot (10 (8 - 6))}{{(8^{2} - 4 \cdot 8 + 20)}^{2}}$

Step 7.2.1.2

Multiply $2$ by $10$ .

$f'' (8) = - \frac{20 (8 - 6)}{{(8^{2} - 4 \cdot 8 + 20)}^{2}}$

Step 7.2.1.3

Subtract $6$ from $8$ .

$f'' (8) = - \frac{20 \cdot 2}{{(8^{2} - 4 \cdot 8 + 20)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f'' (8) = - \frac{20 \cdot 2}{{(64 - 4 \cdot 8 + 20)}^{2}}$

Step 7.2.2.2

Multiply $- 4$ by $8$ .

$f'' (8) = - \frac{20 \cdot 2}{{(64 - 32 + 20)}^{2}}$

Step 7.2.2.3

Subtract $32$ from $64$ .

$f'' (8) = - \frac{20 \cdot 2}{{(32 + 20)}^{2}}$

Step 7.2.2.4

Add $32$ and $20$ .

$f'' (8) = - \frac{20 \cdot 2}{52^{2}}$

Step 7.2.2.5

Raise $52$ to the power of $2$ .

$f'' (8) = - \frac{20 \cdot 2}{2704}$

Step 7.2.3

Reduce the expression by cancelling the common factors.

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Step 7.2.3.1

Multiply $20$ by $2$ .

$f'' (8) = - \frac{40}{2704}$

Step 7.2.3.2

Cancel the common factor of $40$ and $2704$ .

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Step 7.2.3.2.1

Factor $8$ out of $40$ .

$f'' (8) = - \frac{8 (5)}{2704}$

Step 7.2.3.2.2

Cancel the common factors.

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Step 7.2.3.2.2.1

Factor $8$ out of $2704$ .

$f'' (8) = - \frac{8 \cdot 5}{8 \cdot 338}$

Step 7.2.3.2.2.2

Cancel the common factor.

$f'' (8) = - \frac{8 \cdot 5}{8 \cdot 338}$

Step 7.2.3.2.2.3

Rewrite the expression.

$f'' (8) = - \frac{5}{338}$

Step 7.2.4

The final answer is $- \frac{5}{338}$ .

$- \frac{5}{338}$

Step 7.3

The graph is concave down on the interval $(6, \infty)$ because $f'' (8)$ is negative.

Concave down on $(6, \infty)$ since $f'' (x)$ is negative

Step 8

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

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

Concave up on $(- 2, 6)$ since $f'' (x)$ is positive

Concave down on $(6, \infty)$ since $f'' (x)$ is negative

Step 9