Calculus Examples

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

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Multiply $2$ by $2$ .

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

Step 1.1.1.3

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 $g (x) = x - 4$ .

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Step 1.1.1.3.1

To apply the Chain Rule, set $u_{1}$ as $x - 4$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Replace all occurrences of $u_{1}$ with $x - 4$ .

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

Step 1.1.1.4

Differentiate.

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

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

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

Step 1.1.1.4.3

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

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

Step 1.1.1.4.4

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

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

Step 1.1.1.4.5

Simplify the expression.

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Step 1.1.1.4.5.1

Add $1$ and $0$ .

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

Step 1.1.1.4.5.2

Multiply $2$ by $1$ .

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

Step 1.1.1.5

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

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Step 1.1.1.5.1

To apply the Chain Rule, set $u_{2}$ as $x - 2$ .

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

Step 1.1.1.5.2

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

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

Step 1.1.1.5.3

Replace all occurrences of $u_{2}$ with $x - 2$ .

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

Step 1.1.1.6

Differentiate.

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Step 1.1.1.6.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.6.2

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

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

Step 1.1.1.6.3

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

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

Step 1.1.1.6.4

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

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

Step 1.1.1.6.5

Simplify the expression.

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Step 1.1.1.6.5.1

Add $1$ and $0$ .

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

Step 1.1.1.6.5.2

Multiply $x - 2$ by $1$ .

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

Step 1.1.1.7

Simplify.

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Step 1.1.1.7.1

Simplify the numerator.

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Step 1.1.1.7.1.1

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

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Step 1.1.1.7.1.1.1

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

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

Step 1.1.1.7.1.1.2

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

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

Step 1.1.1.7.1.1.3

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

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

Step 1.1.1.7.1.2

Apply the distributive property.

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

Step 1.1.1.7.1.3

Multiply $- 1$ by $- 4$ .

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

Step 1.1.1.7.1.4

Subtract $x$ from $x$ .

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

Step 1.1.1.7.1.5

Subtract $2$ from $0$ .

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

Step 1.1.1.7.1.6

Add $- 2$ and $4$ .

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

Step 1.1.1.7.1.7

Multiply $2$ by $2$ .

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

Step 1.1.1.7.2

Cancel the common factor of $x - 2$ and ${(x - 2)}^{4}$ .

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Step 1.1.1.7.2.1

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

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

Step 1.1.1.7.2.2

Cancel the common factors.

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Step 1.1.1.7.2.2.1

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

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

Step 1.1.1.7.2.2.2

Cancel the common factor.

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

Step 1.1.1.7.2.2.3

Rewrite the expression.

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 (x - 4)}{{(x - 2)}^{3}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x - 4}{{(x - 2)}^{3}}]$ .

$4 \frac{d}{d x} [\frac{x - 4}{{(x - 2)}^{3}}]$

Step 1.1.2.2

$4 \frac{{(x - 2)}^{3} \frac{d}{d x} [x - 4] - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{({(x - 2)}^{3})}^{2}}$

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

Multiply the exponents in ${({(x - 2)}^{3})}^{2}$ .

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Step 1.1.2.3.1.1

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

$4 \frac{{(x - 2)}^{3} \frac{d}{d x} [x - 4] - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{3 \cdot 2}}$

Step 1.1.2.3.1.2

Multiply $3$ by $2$ .

$4 \frac{{(x - 2)}^{3} \frac{d}{d x} [x - 4] - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 1.1.2.3.2

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

$4 \frac{{(x - 2)}^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 1.1.2.3.3

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

$4 \frac{{(x - 2)}^{3} (1 + \frac{d}{d x} [- 4]) - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 1.1.2.3.4

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

$4 \frac{{(x - 2)}^{3} (1 + 0) - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 1.1.2.3.5

Simplify the expression.

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Step 1.1.2.3.5.1

Add $1$ and $0$ .

$4 \frac{{(x - 2)}^{3} \cdot 1 - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 1.1.2.3.5.2

Multiply ${(x - 2)}^{3}$ by $1$ .

$4 \frac{{(x - 2)}^{3} - (x - 4) \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

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 $f (x) = x^{3}$ and $g (x) = x - 2$ .

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Step 1.1.2.4.1

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

$4 \frac{{(x - 2)}^{3} - (x - 4) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

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 $n = 3$ .

$4 \frac{{(x - 2)}^{3} - (x - 4) (3 u^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 1.1.2.4.3

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

$4 \frac{{(x - 2)}^{3} - (x - 4) (3 {(x - 2)}^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 1.1.2.5

Simplify with factoring out.

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Step 1.1.2.5.1

Multiply $3$ by $- 1$ .

$4 \frac{{(x - 2)}^{3} - 3 (x - 4) ({(x - 2)}^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 1.1.2.5.2

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{3} - 3 (x - 4) ({(x - 2)}^{2} \frac{d}{d x} [x - 2])$ .

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Step 1.1.2.5.2.1

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

$4 \frac{{(x - 2)}^{2} (x - 2) - 3 (x - 4) ({(x - 2)}^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 1.1.2.5.2.2

Factor ${(x - 2)}^{2}$ out of $- 3 (x - 4) ({(x - 2)}^{2} \frac{d}{d x} [x - 2])$ .

$4 \frac{{(x - 2)}^{2} (x - 2) + {(x - 2)}^{2} (- 3 (x - 4) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{6}}$

Step 1.1.2.5.2.3

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} (x - 2) + {(x - 2)}^{2} (- 3 (x - 4) (\frac{d}{d x} [x - 2]))$ .

$4 \frac{{(x - 2)}^{2} (x - 2 - 3 (x - 4) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{6}}$

Step 1.1.2.6

Cancel the common factors.

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Step 1.1.2.6.1

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

$4 \frac{{(x - 2)}^{2} (x - 2 - 3 (x - 4) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{2} {(x - 2)}^{4}}$

Step 1.1.2.6.2

Cancel the common factor.

$4 \frac{{(x - 2)}^{2} (x - 2 - 3 (x - 4) (\frac{d}{d x} [x - 2]))}{{(x - 2)}^{2} {(x - 2)}^{4}}$

Step 1.1.2.6.3

Rewrite the expression.

$4 \frac{x - 2 - 3 (x - 4) (\frac{d}{d x} [x - 2])}{{(x - 2)}^{4}}$

Step 1.1.2.7

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

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

Step 1.1.2.8

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

$4 \frac{x - 2 - 3 (x - 4) (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{4}}$

Step 1.1.2.9

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

$4 \frac{x - 2 - 3 (x - 4) (1 + 0)}{{(x - 2)}^{4}}$

Step 1.1.2.10

Combine fractions.

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Step 1.1.2.10.1

Add $1$ and $0$ .

$4 \frac{x - 2 - 3 (x - 4) \cdot 1}{{(x - 2)}^{4}}$

Step 1.1.2.10.2

Multiply $- 3$ by $1$ .

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

Step 1.1.2.10.3

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

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

Step 1.1.2.11

Simplify.

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Step 1.1.2.11.1

Apply the distributive property.

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

Step 1.1.2.11.2

Apply the distributive property.

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

Step 1.1.2.11.3

Simplify the numerator.

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Step 1.1.2.11.3.1

Simplify each term.

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Step 1.1.2.11.3.1.1

Multiply $4$ by $- 2$ .

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

Step 1.1.2.11.3.1.2

Multiply $- 3$ by $4$ .

$\frac{4 x - 8 - 12 x + 4 (- 3 \cdot - 4)}{{(x - 2)}^{4}}$

Step 1.1.2.11.3.1.3

Multiply $4 (- 3 \cdot - 4)$ .

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Step 1.1.2.11.3.1.3.1

Multiply $- 3$ by $- 4$ .

$\frac{4 x - 8 - 12 x + 4 \cdot 12}{{(x - 2)}^{4}}$

Step 1.1.2.11.3.1.3.2

Multiply $4$ by $12$ .

$\frac{4 x - 8 - 12 x + 48}{{(x - 2)}^{4}}$

Step 1.1.2.11.3.2

Subtract $12 x$ from $4 x$ .

$\frac{- 8 x - 8 + 48}{{(x - 2)}^{4}}$

Step 1.1.2.11.3.3

Add $- 8$ and $48$ .

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

Step 1.1.2.11.4

Factor $8$ out of $- 8 x + 40$ .

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Step 1.1.2.11.4.1

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

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

Step 1.1.2.11.4.2

Factor $8$ out of $40$ .

$\frac{8 (- x) + 8 (5)}{{(x - 2)}^{4}}$

Step 1.1.2.11.4.3

Factor $8$ out of $8 (- x) + 8 (5)$ .

$\frac{8 (- x + 5)}{{(x - 2)}^{4}}$

Step 1.1.2.11.5

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

$\frac{8 (- (x) + 5)}{{(x - 2)}^{4}}$

Step 1.1.2.11.6

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{8 (- (x) - 1 (- 5))}{{(x - 2)}^{4}}$

Step 1.1.2.11.7

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{8 (- (x - 5))}{{(x - 2)}^{4}}$

Step 1.1.2.11.8

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{8 (- 1 (x - 5))}{{(x - 2)}^{4}}$

Step 1.1.2.11.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{8 (x - 5)}{{(x - 2)}^{4}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{8 (x - 5)}{{(x - 2)}^{4}}$ .

$- \frac{8 (x - 5)}{{(x - 2)}^{4}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{8 (x - 5)}{{(x - 2)}^{4}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- \frac{8 (x - 5)}{{(x - 2)}^{4}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$8 (x - 5) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $8 (x - 5) = 0$ by $8$ and simplify.

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Step 1.2.3.1.1

Divide each term in $8 (x - 5) = 0$ by $8$ .

$\frac{8 (x - 5)}{8} = \frac{0}{8}$

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 $8$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{8 (x - 5)}{8} = \frac{0}{8}$

Step 1.2.3.1.2.1.2

Divide $x - 5$ by $1$ .

$x - 5 = \frac{0}{8}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $8$ .

$x - 5 = 0$

Step 1.2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2

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

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Step 2.1

Set the denominator in $\frac{{(x - 4)}^{2}}{{(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 2)}^{2} = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 2}$

Step 3

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

$(- \infty, 2) \cup (2, 5) \cup (5, \infty)$

Step 4

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

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Step 4.1

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

$f'' (0) = - \frac{8 ((0) - 5)}{{((0) - 2)}^{4}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $5$ from $0$ .

$f'' (0) = - \frac{8 \cdot - 5}{{(0 - 2)}^{4}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Subtract $2$ from $0$ .

$f'' (0) = - \frac{8 \cdot - 5}{{(- 2)}^{4}}$

Step 4.2.2.2

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

$f'' (0) = - \frac{8 \cdot - 5}{16}$

Step 4.2.3

Reduce the expression by cancelling the common factors.

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Step 4.2.3.1

Multiply $8$ by $- 5$ .

$f'' (0) = - \frac{- 40}{16}$

Step 4.2.3.2

Cancel the common factor of $- 40$ and $16$ .

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Step 4.2.3.2.1

Factor $8$ out of $- 40$ .

$f'' (0) = - \frac{8 (- 5)}{16}$

Step 4.2.3.2.2

Cancel the common factors.

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Step 4.2.3.2.2.1

Factor $8$ out of $16$ .

$f'' (0) = - \frac{8 \cdot - 5}{8 \cdot 2}$

Step 4.2.3.2.2.2

Cancel the common factor.

$f'' (0) = - \frac{8 \cdot - 5}{8 \cdot 2}$

Step 4.2.3.2.2.3

Rewrite the expression.

$f'' (0) = - \frac{- 5}{2}$

Step 4.2.3.3

Move the negative in front of the fraction.

$f'' (0) = \frac{5}{2}$

Step 4.2.4

The final answer is $\frac{5}{2}$ .

$\frac{5}{2}$

Step 4.3

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

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

Step 5

Substitute any number from the interval $(2, 5)$ 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.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Subtract $5$ from $4$ .

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

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Subtract $2$ from $4$ .

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

Step 5.2.2.2

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

$f'' (4) = - \frac{8 \cdot - 1}{16}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Multiply $8$ by $- 1$ .

$f'' (4) = - \frac{- 8}{16}$

Step 5.2.3.2

Cancel the common factor of $- 8$ and $16$ .

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Step 5.2.3.2.1

Factor $8$ out of $- 8$ .

$f'' (4) = - \frac{8 (- 1)}{16}$

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 $16$ .

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

Step 5.2.3.2.2.2

Cancel the common factor.

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

Step 5.2.3.2.2.3

Rewrite the expression.

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

Step 5.2.3.3

Move the negative in front of the fraction.

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

Step 5.2.4

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

$\frac{1}{2}$

Step 5.3

The graph is concave up on the interval $(2, 5)$ because $f'' (4)$ is positive.

Concave up on $(2, 5)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(5, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

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

Step 6.2

Simplify the result.

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Step 6.2.1

Subtract $5$ from $8$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Subtract $2$ from $8$ .

$f'' (8) = - \frac{8 \cdot 3}{6^{4}}$

Step 6.2.2.2

Raise $6$ to the power of $4$ .

$f'' (8) = - \frac{8 \cdot 3}{1296}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $8$ by $3$ .

$f'' (8) = - \frac{24}{1296}$

Step 6.2.3.2

Cancel the common factor of $24$ and $1296$ .

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Step 6.2.3.2.1

Factor $24$ out of $24$ .

$f'' (8) = - \frac{24 (1)}{1296}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $24$ out of $1296$ .

$f'' (8) = - \frac{24 \cdot 1}{24 \cdot 54}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (8) = - \frac{24 \cdot 1}{24 \cdot 54}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (8) = - \frac{1}{54}$

Step 6.2.4

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

$- \frac{1}{54}$

Step 6.3

The graph is concave down on the interval $(5, \infty)$ because $f'' (8)$ is negative.

Concave down on $(5, \infty)$ since $f'' (x)$ is negative

Step 7

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

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

Concave up on $(2, 5)$ since $f'' (x)$ is positive

Concave down on $(5, \infty)$ since $f'' (x)$ is negative

Step 8