Calculus Examples

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

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.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $1 + \frac{1}{x} + \frac{7}{x^{2}} + \frac{1}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{7}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 1.1.1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 1.1.1.2.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

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 $n = - 1$ .

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

Step 1.1.1.3

Evaluate $\frac{d}{d x} [\frac{7}{x^{2}}]$ .

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

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

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Step 1.1.1.3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 1.1.1.3.3.2

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

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

Step 1.1.1.3.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 1.1.1.3.4

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

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

Step 1.1.1.3.5

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

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Step 1.1.1.3.5.1

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

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

Step 1.1.1.3.5.2

Multiply $2$ by $- 2$ .

$f' (x) = 0 - x^{- 2} + 7 (- x^{- 4} (2 x)) + \frac{d}{d x} (\frac{1}{x^{3}})$

Step 1.1.1.3.6

Multiply $2$ by $- 1$ .

$f' (x) = 0 - x^{- 2} + 7 (- 2 x^{- 4} x) + \frac{d}{d x} (\frac{1}{x^{3}})$

Step 1.1.1.3.7

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

$f' (x) = 0 - x^{- 2} + 7 (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (\frac{1}{x^{3}})$

Step 1.1.1.3.8

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

$f' (x) = 0 - x^{- 2} + 7 (- 2 x^{1 - 4}) + \frac{d}{d x} (\frac{1}{x^{3}})$

Step 1.1.1.3.9

Subtract $4$ from $1$ .

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

Step 1.1.1.3.10

Multiply $- 2$ by $7$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} + \frac{d}{d x} (\frac{1}{x^{3}})$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [\frac{1}{x^{3}}]$ .

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Step 1.1.1.4.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} + \frac{d}{d x} ({(x^{3})}^{- 1})$

Step 1.1.1.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 1.1.1.4.2.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} + \frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} (x^{3})$

Step 1.1.1.4.2.2

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

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - {u_{2}}^{- 2} \frac{d}{d x} x^{3}$

Step 1.1.1.4.2.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - {(x^{3})}^{- 2} \frac{d}{d x} x^{3}$

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 $n = 3$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - {(x^{3})}^{- 2} (3 x^{2})$

Step 1.1.1.4.4

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

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Step 1.1.1.4.4.1

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

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - x^{3 \cdot - 2} (3 x^{2})$

Step 1.1.1.4.4.2

Multiply $3$ by $- 2$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - x^{- 6} (3 x^{2})$

Step 1.1.1.4.5

Multiply $3$ by $- 1$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - 3 x^{- 6} x^{2}$

Step 1.1.1.4.6

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 1.1.1.4.6.1

Move $x^{2}$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - 3 (x^{2} x^{- 6})$

Step 1.1.1.4.6.2

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

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - 3 x^{2 - 6}$

Step 1.1.1.4.6.3

Subtract $6$ from $2$ .

$f' (x) = 0 - x^{- 2} - 14 x^{- 3} - 3 x^{- 4}$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

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

$f' (x) = 0 - \frac{1}{x^{2}} - 14 x^{- 3} - 3 x^{- 4}$

Step 1.1.1.5.2

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

$f' (x) = 0 - \frac{1}{x^{2}} - 14 \frac{1}{x^{3}} - 3 x^{- 4}$

Step 1.1.1.5.3

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

$f' (x) = 0 - \frac{1}{x^{2}} - 14 \frac{1}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.1.5.4

Combine terms.

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Step 1.1.1.5.4.1

Subtract $\frac{1}{x^{2}}$ from $0$ .

$f' (x) = - \frac{1}{x^{2}} - 14 \frac{1}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.1.5.4.2

Combine $- 14$ and $\frac{1}{x^{3}}$ .

$f' (x) = - \frac{1}{x^{2}} + \frac{- 14}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.1.5.4.3

Move the negative in front of the fraction.

$f' (x) = - \frac{1}{x^{2}} - \frac{14}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.1.5.4.4

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$f' (x) = - \frac{1}{x^{2}} - \frac{14}{x^{3}} + \frac{- 3}{x^{4}}$

Step 1.1.1.5.4.5

Move the negative in front of the fraction.

$f' (x) = - \frac{1}{x^{2}} - \frac{14}{x^{3}} - \frac{3}{x^{4}}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- \frac{1}{x^{2}} - \frac{14}{x^{3}} - \frac{3}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{14}{x^{3}}] + \frac{d}{d x} [- \frac{3}{x^{4}}]$ .

$f'' (x) = \frac{d}{d x} (- \frac{1}{x^{2}}) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x^{2}}]$ .

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Step 1.1.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{1}{x^{2}} + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.2

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

$f'' (x) = - \frac{d}{d x} {(x^{2})}^{- 1} + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.3

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

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Step 1.1.2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$f'' (x) = - (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x^{2})) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.3.2

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

$f'' (x) = - (- {u_{1}}^{- 2} \frac{d}{d x} x^{2}) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$f'' (x) = - (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.4

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

$f'' (x) = - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot \frac{d}{d x} (- 1) + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.6

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

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Step 1.1.2.2.6.1

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

$f'' (x) = - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.6.2

Multiply $2$ by $- 2$ .

$f'' (x) = - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.7

Multiply $2$ by $- 1$ .

$f'' (x) = - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.8

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

$f'' (x) = - (- 2 (x \cdot x^{- 4})) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.9

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

$f'' (x) = - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.10

Subtract $4$ from $1$ .

$f'' (x) = - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.11

Multiply $- 2$ by $- 1$ .

$f'' (x) = 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$f'' (x) = 2 x^{- 3} + 0 + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.2.13

Add $2 x^{- 3}$ and $0$ .

$f'' (x) = 2 x^{- 3} + \frac{d}{d x} (- \frac{14}{x^{3}}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- \frac{14}{x^{3}}]$ .

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Step 1.1.2.3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- \frac{14}{x^{3}}$ with respect to $x$ is $- 14 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 2 x^{- 3} - 14 \frac{d}{d x} \frac{1}{x^{3}} + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f'' (x) = 2 x^{- 3} - 14 \frac{d}{d x} {(x^{3})}^{- 1} + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 1.1.2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

$f'' (x) = 2 x^{- 3} - 14 (\frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.3.2

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

$f'' (x) = 2 x^{- 3} - 14 (- {u_{2}}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

$f'' (x) = 2 x^{- 3} - 14 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 2 x^{- 3} - 14 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.5

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

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Step 1.1.2.3.5.1

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

$f'' (x) = 2 x^{- 3} - 14 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 2 x^{- 3} - 14 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.6

Multiply $3$ by $- 1$ .

$f'' (x) = 2 x^{- 3} - 14 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 1.1.2.3.7.1

Move $x^{2}$ .

$f'' (x) = 2 x^{- 3} - 14 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.7.2

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

$f'' (x) = 2 x^{- 3} - 14 (- 3 x^{2 - 6}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.7.3

Subtract $6$ from $2$ .

$f'' (x) = 2 x^{- 3} - 14 (- 3 x^{- 4}) + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.3.8

Multiply $- 3$ by $- 14$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} + \frac{d}{d x} (- \frac{3}{x^{4}})$

Step 1.1.2.4

Evaluate $\frac{d}{d x} [- \frac{3}{x^{4}}]$ .

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Step 1.1.2.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x^{4}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 \frac{d}{d x} \frac{1}{x^{4}}$

Step 1.1.2.4.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 \frac{d}{d x} {(x^{4})}^{- 1}$

Step 1.1.2.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4}$ .

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Step 1.1.2.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{4}$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (\frac{d}{d u (3)} ({u_{3}}^{- 1}) \frac{d}{d x} (x^{4}))$

Step 1.1.2.4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = - 1$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- {u_{3}}^{- 2} \frac{d}{d x} x^{4})$

Step 1.1.2.4.3.3

Replace all occurrences of $u_{3}$ with $x^{4}$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- {(x^{4})}^{- 2} \frac{d}{d x} x^{4})$

Step 1.1.2.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 1.1.2.4.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 1.1.2.4.5.1

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

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- x^{4 \cdot - 2} (4 x^{3}))$

Step 1.1.2.4.5.2

Multiply $4$ by $- 2$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- x^{- 8} (4 x^{3}))$

Step 1.1.2.4.6

Multiply $4$ by $- 1$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- 4 x^{- 8} x^{3})$

Step 1.1.2.4.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.4.7.1

Move $x^{3}$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- 4 (x^{3} x^{- 8}))$

Step 1.1.2.4.7.2

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

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- 4 x^{3 - 8})$

Step 1.1.2.4.7.3

Subtract $8$ from $3$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} - 3 (- 4 x^{- 5})$

Step 1.1.2.4.8

Multiply $- 4$ by $- 3$ .

$f'' (x) = 2 x^{- 3} + 42 x^{- 4} + 12 x^{- 5}$

Step 1.1.2.5

Simplify.

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Step 1.1.2.5.1

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

$f'' (x) = 2 (\frac{1}{x^{3}}) + 42 x^{- 4} + 12 x^{- 5}$

Step 1.1.2.5.2

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

$f'' (x) = 2 (\frac{1}{x^{3}}) + 42 (\frac{1}{x^{4}}) + 12 x^{- 5}$

Step 1.1.2.5.3

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

$f'' (x) = 2 (\frac{1}{x^{3}}) + 42 (\frac{1}{x^{4}}) + 12 (\frac{1}{x^{5}})$

Step 1.1.2.5.4

Combine terms.

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Step 1.1.2.5.4.1

Combine $2$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{2}{x^{3}} + 42 (\frac{1}{x^{4}}) + 12 (\frac{1}{x^{5}})$

Step 1.1.2.5.4.2

Combine $42$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{2}{x^{3}} + \frac{42}{x^{4}} + 12 (\frac{1}{x^{5}})$

Step 1.1.2.5.4.3

Combine $12$ and $\frac{1}{x^{5}}$ .

$f'' (x) = \frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}}$ .

$\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{3}, x^{4}, x^{5}, 1$

Step 1.2.2.2

Since $x^{3}, x^{4}, x^{5}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{3}, x^{4}, x^{5}$ .

Step 1.2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.2.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.2.2.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.2.2.6

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

Step 1.2.2.7

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 1.2.2.8

The factors for $x^{5}$ are $x \cdot x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $5$ times.

$x^{5} = x \cdot x \cdot x \cdot x \cdot x$

$x$ occurs $5$ times.

Step 1.2.2.9

The LCM of $x^{3}, x^{4}, x^{5}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x \cdot x$

Step 1.2.2.10

Simplify $x \cdot x \cdot x \cdot x \cdot x$ .

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Step 1.2.2.10.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x \cdot x$

Step 1.2.2.10.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.2.10.2.1

Multiply $x^{2}$ by $x$ .

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Step 1.2.2.10.2.1.1

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

$x^{2} \cdot x^{1} \cdot x \cdot x$

Step 1.2.2.10.2.1.2

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

$x^{2 + 1} \cdot x \cdot x$

Step 1.2.2.10.2.2

Add $2$ and $1$ .

$x^{3} \cdot x \cdot x$

Step 1.2.2.10.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.2.2.10.3.1

Multiply $x^{3}$ by $x$ .

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Step 1.2.2.10.3.1.1

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

$x^{3} \cdot x^{1} \cdot x$

Step 1.2.2.10.3.1.2

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

$x^{3 + 1} \cdot x$

Step 1.2.2.10.3.2

Add $3$ and $1$ .

$x^{4} \cdot x$

Step 1.2.2.10.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.2.2.10.4.1

Multiply $x^{4}$ by $x$ .

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Step 1.2.2.10.4.1.1

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

$x^{4} \cdot x^{1}$

Step 1.2.2.10.4.1.2

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

$x^{4 + 1}$

Step 1.2.2.10.4.2

Add $4$ and $1$ .

$x^{5}$

Step 1.2.3

Multiply each term in $\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}} = 0$ by $x^{5}$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $\frac{2}{x^{3}} + \frac{42}{x^{4}} + \frac{12}{x^{5}} = 0$ by $x^{5}$ .

$\frac{2}{x^{3}} x^{5} + \frac{42}{x^{4}} x^{5} + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Cancel the common factor of $x^{3}$ .

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Step 1.2.3.2.1.1.1

Factor $x^{3}$ out of $x^{5}$ .

$\frac{2}{x^{3}} (x^{3} x^{2}) + \frac{42}{x^{4}} x^{5} + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.1.2

Cancel the common factor.

$\frac{2}{x^{3}} (x^{3} x^{2}) + \frac{42}{x^{4}} x^{5} + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.1.3

Rewrite the expression.

$2 x^{2} + \frac{42}{x^{4}} x^{5} + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.2

Cancel the common factor of $x^{4}$ .

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Step 1.2.3.2.1.2.1

Factor $x^{4}$ out of $x^{5}$ .

$2 x^{2} + \frac{42}{x^{4}} (x^{4} x) + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$2 x^{2} + \frac{42}{x^{4}} (x^{4} x) + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$2 x^{2} + 42 x + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.3

Cancel the common factor of $x^{5}$ .

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Step 1.2.3.2.1.3.1

Cancel the common factor.

$2 x^{2} + 42 x + \frac{12}{x^{5}} x^{5} = 0 x^{5}$

Step 1.2.3.2.1.3.2

Rewrite the expression.

$2 x^{2} + 42 x + 12 = 0 x^{5}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $x^{5}$ .

$2 x^{2} + 42 x + 12 = 0$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Factor $2$ out of $2 x^{2} + 42 x + 12$ .

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Step 1.2.4.1.1

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

$2 (x^{2}) + 42 x + 12 = 0$

Step 1.2.4.1.2

Factor $2$ out of $42 x$ .

$2 (x^{2}) + 2 (21 x) + 12 = 0$

Step 1.2.4.1.3

Factor $2$ out of $12$ .

$2 x^{2} + 2 (21 x) + 2 \cdot 6 = 0$

Step 1.2.4.1.4

Factor $2$ out of $2 x^{2} + 2 (21 x)$ .

$2 (x^{2} + 21 x) + 2 \cdot 6 = 0$

Step 1.2.4.1.5

Factor $2$ out of $2 (x^{2} + 21 x) + 2 \cdot 6$ .

$2 (x^{2} + 21 x + 6) = 0$

Step 1.2.4.2

Divide each term in $2 (x^{2} + 21 x + 6) = 0$ by $2$ and simplify.

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Step 1.2.4.2.1

Divide each term in $2 (x^{2} + 21 x + 6) = 0$ by $2$ .

$\frac{2 (x^{2} + 21 x + 6)}{2} = \frac{0}{2}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 21 x + 6)}{2} = \frac{0}{2}$

Step 1.2.4.2.2.1.2

Divide $x^{2} + 21 x + 6$ by $1$ .

$x^{2} + 21 x + 6 = \frac{0}{2}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Divide $0$ by $2$ .

$x^{2} + 21 x + 6 = 0$

Step 1.2.4.3

Use the quadratic formula to find the solutions.

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

Step 1.2.4.4

Substitute the values $a = 1$ , $b = 21$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 21 \pm \sqrt{21^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

Step 1.2.4.5

Simplify.

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Step 1.2.4.5.1

Simplify the numerator.

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Step 1.2.4.5.1.1

Raise $21$ to the power of $2$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.5.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.5.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 21 \pm \sqrt{441 - 24}}{2 \cdot 1}$

Step 1.2.4.5.1.3

Subtract $24$ from $441$ .

$x = \frac{- 21 \pm \sqrt{417}}{2 \cdot 1}$

Step 1.2.4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 21 \pm \sqrt{417}}{2}$

Step 1.2.4.6

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

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Step 1.2.4.6.1

Simplify the numerator.

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Step 1.2.4.6.1.1

Raise $21$ to the power of $2$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.6.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.6.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 21 \pm \sqrt{441 - 24}}{2 \cdot 1}$

Step 1.2.4.6.1.3

Subtract $24$ from $441$ .

$x = \frac{- 21 \pm \sqrt{417}}{2 \cdot 1}$

Step 1.2.4.6.2

Multiply $2$ by $1$ .

$x = \frac{- 21 \pm \sqrt{417}}{2}$

Step 1.2.4.6.3

Change the $\pm$ to $+$ .

$x = \frac{- 21 + \sqrt{417}}{2}$

Step 1.2.4.6.4

Rewrite $- 21$ as $- 1 (21)$ .

$x = \frac{- 1 \cdot 21 + \sqrt{417}}{2}$

Step 1.2.4.6.5

Factor $- 1$ out of $\sqrt{417}$ .

$x = \frac{- 1 \cdot 21 - 1 (- \sqrt{417})}{2}$

Step 1.2.4.6.6

Factor $- 1$ out of $- 1 (21) - 1 (- \sqrt{417})$ .

$x = \frac{- 1 (21 - \sqrt{417})}{2}$

Step 1.2.4.6.7

Move the negative in front of the fraction.

$x = - \frac{21 - \sqrt{417}}{2}$

Step 1.2.4.7

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

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Step 1.2.4.7.1

Simplify the numerator.

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Step 1.2.4.7.1.1

Raise $21$ to the power of $2$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.7.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.4.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 21 \pm \sqrt{441 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.4.7.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 21 \pm \sqrt{441 - 24}}{2 \cdot 1}$

Step 1.2.4.7.1.3

Subtract $24$ from $441$ .

$x = \frac{- 21 \pm \sqrt{417}}{2 \cdot 1}$

Step 1.2.4.7.2

Multiply $2$ by $1$ .

$x = \frac{- 21 \pm \sqrt{417}}{2}$

Step 1.2.4.7.3

Change the $\pm$ to $-$ .

$x = \frac{- 21 - \sqrt{417}}{2}$

Step 1.2.4.7.4

Rewrite $- 21$ as $- 1 (21)$ .

$x = \frac{- 1 \cdot 21 - \sqrt{417}}{2}$

Step 1.2.4.7.5

Factor $- 1$ out of $- \sqrt{417}$ .

$x = \frac{- 1 \cdot 21 - (\sqrt{417})}{2}$

Step 1.2.4.7.6

Factor $- 1$ out of $- 1 (21) - (\sqrt{417})$ .

$x = \frac{- 1 (21 + \sqrt{417})}{2}$

Step 1.2.4.7.7

Move the negative in front of the fraction.

$x = - \frac{21 + \sqrt{417}}{2}$

Step 1.2.4.8

The final answer is the combination of both solutions.

$x = - \frac{21 - \sqrt{417}}{2}, - \frac{21 + \sqrt{417}}{2}$

Step 2

Find the domain of $f (x) = 1 + \frac{1}{x} + \frac{7}{x^{2}} + \frac{1}{x^{3}}$ .

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Step 2.1

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

$x = 0$

Step 2.2

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

$x^{2} = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.3.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.2.2

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

$x = \pm 0$

Step 2.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

Set the denominator in $\frac{1}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 2.5.2

Simplify $\sqrt[3]{0}$ .

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Step 2.5.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.5.2.2

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

$x = 0$

Step 2.6

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 3

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

$(- \infty, - \frac{21 + \sqrt{417}}{2}) \cup (- \frac{21 + \sqrt{417}}{2}, - \frac{21 - \sqrt{417}}{2}) \cup (- \frac{21 - \sqrt{417}}{2}, 0) \cup (0, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{21 + \sqrt{417}}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 23$ in the expression.

$f'' (- 23) = \frac{2}{{(- 23)}^{3}} + \frac{42}{{(- 23)}^{4}} + \frac{12}{{(- 23)}^{5}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Find the common denominator.

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Step 4.2.1.1

Raise $- 23$ to the power of $3$ .

$f'' (- 23) = \frac{2}{- 12167} + \frac{42}{{(- 23)}^{4}} + \frac{12}{{(- 23)}^{5}}$

Step 4.2.1.2

Raise $- 23$ to the power of $4$ .

$f'' (- 23) = \frac{2}{- 12167} + \frac{42}{279841} + \frac{12}{{(- 23)}^{5}}$

Step 4.2.1.3

Raise $- 23$ to the power of $5$ .

$f'' (- 23) = \frac{2}{- 12167} + \frac{42}{279841} + \frac{12}{- 6436343}$

Step 4.2.1.4

Multiply $\frac{2}{- 12167}$ by $\frac{- 529}{- 529}$ .

$f'' (- 23) = \frac{2}{- 12167} \cdot \frac{- 529}{- 529} + \frac{42}{279841} + \frac{12}{- 6436343}$

Step 4.2.1.5

Multiply $\frac{2}{- 12167}$ by $\frac{- 529}{- 529}$ .

$f'' (- 23) = \frac{2 \cdot - 529}{- 12167 \cdot - 529} + \frac{42}{279841} + \frac{12}{- 6436343}$

Step 4.2.1.6

Multiply $\frac{42}{279841}$ by $\frac{23}{23}$ .

$f'' (- 23) = \frac{2 \cdot - 529}{- 12167 \cdot - 529} + \frac{42}{279841} \cdot \frac{23}{23} + \frac{12}{- 6436343}$

Step 4.2.1.7

Multiply $\frac{42}{279841}$ by $\frac{23}{23}$ .

$f'' (- 23) = \frac{2 \cdot - 529}{- 12167 \cdot - 529} + \frac{42 \cdot 23}{279841 \cdot 23} + \frac{12}{- 6436343}$

Step 4.2.1.8

Multiply $\frac{12}{- 6436343}$ by $\frac{- 1}{- 1}$ .

$f'' (- 23) = \frac{2 \cdot - 529}{- 12167 \cdot - 529} + \frac{42 \cdot 23}{279841 \cdot 23} + \frac{12}{- 6436343} \cdot \frac{- 1}{- 1}$

Step 4.2.1.9

Multiply $\frac{12}{- 6436343}$ by $\frac{- 1}{- 1}$ .

$f'' (- 23) = \frac{2 \cdot - 529}{- 12167 \cdot - 529} + \frac{42 \cdot 23}{279841 \cdot 23} + \frac{12 \cdot - 1}{- 6436343 \cdot - 1}$

Step 4.2.1.10

Multiply $- 12167$ by $- 529$ .

$f'' (- 23) = \frac{2 \cdot - 529}{6436343} + \frac{42 \cdot 23}{279841 \cdot 23} + \frac{12 \cdot - 1}{- 6436343 \cdot - 1}$

Step 4.2.1.11

Reorder the factors of $279841 \cdot 23$ .

$f'' (- 23) = \frac{2 \cdot - 529}{6436343} + \frac{42 \cdot 23}{23 \cdot 279841} + \frac{12 \cdot - 1}{- 6436343 \cdot - 1}$

Step 4.2.1.12

Multiply $23$ by $279841$ .

$f'' (- 23) = \frac{2 \cdot - 529}{6436343} + \frac{42 \cdot 23}{6436343} + \frac{12 \cdot - 1}{- 6436343 \cdot - 1}$

Step 4.2.1.13

Multiply $- 6436343$ by $- 1$ .

$f'' (- 23) = \frac{2 \cdot - 529}{6436343} + \frac{42 \cdot 23}{6436343} + \frac{12 \cdot - 1}{6436343}$

Step 4.2.2

Combine the numerators over the common denominator.

$f'' (- 23) = \frac{2 \cdot - 529 + 42 \cdot 23 + 12 \cdot - 1}{6436343}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply $2$ by $- 529$ .

$f'' (- 23) = \frac{- 1058 + 42 \cdot 23 + 12 \cdot - 1}{6436343}$

Step 4.2.3.2

Multiply $42$ by $23$ .

$f'' (- 23) = \frac{- 1058 + 966 + 12 \cdot - 1}{6436343}$

Step 4.2.3.3

Multiply $12$ by $- 1$ .

$f'' (- 23) = \frac{- 1058 + 966 - 12}{6436343}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Add $- 1058$ and $966$ .

$f'' (- 23) = \frac{- 92 - 12}{6436343}$

Step 4.2.4.2

Subtract $12$ from $- 92$ .

$f'' (- 23) = \frac{- 104}{6436343}$

Step 4.2.4.3

Move the negative in front of the fraction.

$f'' (- 23) = - \frac{104}{6436343}$

Step 4.2.5

The final answer is $- \frac{104}{6436343}$ .

$- \frac{104}{6436343}$

Step 4.3

The graph is concave down on the interval $(- \infty, - \frac{21 + \sqrt{417}}{2})$ because $f'' (- 23)$ is negative.

Concave down on $(- \infty, - \frac{21 + \sqrt{417}}{2})$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- \frac{21 + \sqrt{417}}{2}, - \frac{21 - \sqrt{417}}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

$f'' (- 3) = \frac{2}{{(- 3)}^{3}} + \frac{42}{{(- 3)}^{4}} + \frac{12}{{(- 3)}^{5}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

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

$f'' (- 3) = \frac{2}{- 27} + \frac{42}{{(- 3)}^{4}} + \frac{12}{{(- 3)}^{5}}$

Step 5.2.1.2

Raise $- 3$ to the power of $4$ .

$f'' (- 3) = \frac{2}{- 27} + \frac{42}{81} + \frac{12}{{(- 3)}^{5}}$

Step 5.2.1.3

Raise $- 3$ to the power of $5$ .

$f'' (- 3) = \frac{2}{- 27} + \frac{42}{81} + \frac{12}{- 243}$

Step 5.2.1.4

Multiply $\frac{2}{- 27}$ by $\frac{- 9}{- 9}$ .

$f'' (- 3) = \frac{2}{- 27} \cdot \frac{- 9}{- 9} + \frac{42}{81} + \frac{12}{- 243}$

Step 5.2.1.5

Multiply $\frac{2}{- 27}$ by $\frac{- 9}{- 9}$ .

$f'' (- 3) = \frac{2 \cdot - 9}{- 27 \cdot - 9} + \frac{42}{81} + \frac{12}{- 243}$

Step 5.2.1.6

Multiply $\frac{42}{81}$ by $\frac{3}{3}$ .

$f'' (- 3) = \frac{2 \cdot - 9}{- 27 \cdot - 9} + \frac{42}{81} \cdot \frac{3}{3} + \frac{12}{- 243}$

Step 5.2.1.7

Multiply $\frac{42}{81}$ by $\frac{3}{3}$ .

$f'' (- 3) = \frac{2 \cdot - 9}{- 27 \cdot - 9} + \frac{42 \cdot 3}{81 \cdot 3} + \frac{12}{- 243}$

Step 5.2.1.8

Multiply $\frac{12}{- 243}$ by $\frac{- 1}{- 1}$ .

$f'' (- 3) = \frac{2 \cdot - 9}{- 27 \cdot - 9} + \frac{42 \cdot 3}{81 \cdot 3} + \frac{12}{- 243} \cdot \frac{- 1}{- 1}$

Step 5.2.1.9

Multiply $\frac{12}{- 243}$ by $\frac{- 1}{- 1}$ .

$f'' (- 3) = \frac{2 \cdot - 9}{- 27 \cdot - 9} + \frac{42 \cdot 3}{81 \cdot 3} + \frac{12 \cdot - 1}{- 243 \cdot - 1}$

Step 5.2.1.10

Multiply $- 27$ by $- 9$ .

$f'' (- 3) = \frac{2 \cdot - 9}{243} + \frac{42 \cdot 3}{81 \cdot 3} + \frac{12 \cdot - 1}{- 243 \cdot - 1}$

Step 5.2.1.11

Reorder the factors of $81 \cdot 3$ .

$f'' (- 3) = \frac{2 \cdot - 9}{243} + \frac{42 \cdot 3}{3 \cdot 81} + \frac{12 \cdot - 1}{- 243 \cdot - 1}$

Step 5.2.1.12

Multiply $3$ by $81$ .

$f'' (- 3) = \frac{2 \cdot - 9}{243} + \frac{42 \cdot 3}{243} + \frac{12 \cdot - 1}{- 243 \cdot - 1}$

Step 5.2.1.13

Multiply $- 243$ by $- 1$ .

$f'' (- 3) = \frac{2 \cdot - 9}{243} + \frac{42 \cdot 3}{243} + \frac{12 \cdot - 1}{243}$

Step 5.2.2

Combine the numerators over the common denominator.

$f'' (- 3) = \frac{2 \cdot - 9 + 42 \cdot 3 + 12 \cdot - 1}{243}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply $2$ by $- 9$ .

$f'' (- 3) = \frac{- 18 + 42 \cdot 3 + 12 \cdot - 1}{243}$

Step 5.2.3.2

Multiply $42$ by $3$ .

$f'' (- 3) = \frac{- 18 + 126 + 12 \cdot - 1}{243}$

Step 5.2.3.3

Multiply $12$ by $- 1$ .

$f'' (- 3) = \frac{- 18 + 126 - 12}{243}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Add $- 18$ and $126$ .

$f'' (- 3) = \frac{108 - 12}{243}$

Step 5.2.4.2

Subtract $12$ from $108$ .

$f'' (- 3) = \frac{96}{243}$

Step 5.2.4.3

Cancel the common factor of $96$ and $243$ .

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Step 5.2.4.3.1

Factor $3$ out of $96$ .

$f'' (- 3) = \frac{3 (32)}{243}$

Step 5.2.4.3.2

Cancel the common factors.

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Step 5.2.4.3.2.1

Factor $3$ out of $243$ .

$f'' (- 3) = \frac{3 \cdot 32}{3 \cdot 81}$

Step 5.2.4.3.2.2

Cancel the common factor.

$f'' (- 3) = \frac{3 \cdot 32}{3 \cdot 81}$

Step 5.2.4.3.2.3

Rewrite the expression.

$f'' (- 3) = \frac{32}{81}$

Step 5.2.5

The final answer is $\frac{32}{81}$ .

$\frac{32}{81}$

Step 5.3

The graph is concave up on the interval $(- \frac{21 + \sqrt{417}}{2}, - \frac{21 - \sqrt{417}}{2})$ because $f'' (- 3)$ is positive.

Concave up on $(- \frac{21 + \sqrt{417}}{2}, - \frac{21 - \sqrt{417}}{2})$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- \frac{21 - \sqrt{417}}{2}, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 0.14$ in the expression.

$f'' (- 0.14) = \frac{2}{{(- 0.14)}^{3}} + \frac{42}{{(- 0.14)}^{4}} + \frac{12}{{(- 0.14)}^{5}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 0.14$ to the power of $3$ .

$f'' (- 0.14) = \frac{2}{- 0.002744} + \frac{42}{{(- 0.14)}^{4}} + \frac{12}{{(- 0.14)}^{5}}$

Step 6.2.1.2

Divide $2$ by $- 0.002744$ .

$f'' (- 0.14) = - 728.86297376 + \frac{42}{{(- 0.14)}^{4}} + \frac{12}{{(- 0.14)}^{5}}$

Step 6.2.1.3

Raise $- 0.14$ to the power of $4$ .

$f'' (- 0.14) = - 728.86297376 + \frac{42}{0.00038416} + \frac{12}{{(- 0.14)}^{5}}$

Step 6.2.1.4

Divide $42$ by $0.00038416$ .

$f'' (- 0.14) = - 728.86297376 + 109329.44606413 + \frac{12}{{(- 0.14)}^{5}}$

Step 6.2.1.5

Raise $- 0.14$ to the power of $5$ .

$f'' (- 0.14) = - 728.86297376 + 109329.44606413 + \frac{12}{- 0.00005378}$

Step 6.2.1.6

Divide $12$ by $- 0.00005378$ .

$f'' (- 0.14) = - 728.86297376 + 109329.44606413 - 223121.31849824$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 728.86297376$ and $109329.44606413$ .

$f'' (- 0.14) = 108600.58309037 - 223121.31849824$

Step 6.2.2.2

Subtract $223121.31849824$ from $108600.58309037$ .

$f'' (- 0.14) = - 114520.73540786$

Step 6.2.3

The final answer is $- 114520.73540786$ .

$- 114520.73540786$

Step 6.3

The graph is concave down on the interval $(- \frac{21 - \sqrt{417}}{2}, 0)$ because $f'' (- 0.14)$ is negative.

Concave down on $(- \frac{21 - \sqrt{417}}{2}, 0)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = \frac{2}{{(2)}^{3}} + \frac{42}{{(2)}^{4}} + \frac{12}{{(2)}^{5}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Find the common denominator.

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Step 7.2.1.1

Raise $2$ to the power of $3$ .

$f'' (2) = \frac{2}{8} + \frac{42}{{(2)}^{4}} + \frac{12}{{(2)}^{5}}$

Step 7.2.1.2

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

$f'' (2) = \frac{2}{8} + \frac{42}{16} + \frac{12}{{(2)}^{5}}$

Step 7.2.1.3

Raise $2$ to the power of $5$ .

$f'' (2) = \frac{2}{8} + \frac{42}{16} + \frac{12}{32}$

Step 7.2.1.4

Multiply $\frac{2}{8}$ by $\frac{4}{4}$ .

$f'' (2) = \frac{2}{8} \cdot \frac{4}{4} + \frac{42}{16} + \frac{12}{32}$

Step 7.2.1.5

Multiply $\frac{2}{8}$ by $\frac{4}{4}$ .

$f'' (2) = \frac{2 \cdot 4}{8 \cdot 4} + \frac{42}{16} + \frac{12}{32}$

Step 7.2.1.6

Multiply $\frac{42}{16}$ by $\frac{2}{2}$ .

$f'' (2) = \frac{2 \cdot 4}{8 \cdot 4} + \frac{42}{16} \cdot \frac{2}{2} + \frac{12}{32}$

Step 7.2.1.7

Multiply $\frac{42}{16}$ by $\frac{2}{2}$ .

$f'' (2) = \frac{2 \cdot 4}{8 \cdot 4} + \frac{42 \cdot 2}{16 \cdot 2} + \frac{12}{32}$

Step 7.2.1.8

Reorder the factors of $8 \cdot 4$ .

$f'' (2) = \frac{2 \cdot 4}{4 \cdot 8} + \frac{42 \cdot 2}{16 \cdot 2} + \frac{12}{32}$

Step 7.2.1.9

Multiply $4$ by $8$ .

$f'' (2) = \frac{2 \cdot 4}{32} + \frac{42 \cdot 2}{16 \cdot 2} + \frac{12}{32}$

Step 7.2.1.10

Reorder the factors of $16 \cdot 2$ .

$f'' (2) = \frac{2 \cdot 4}{32} + \frac{42 \cdot 2}{2 \cdot 16} + \frac{12}{32}$

Step 7.2.1.11

Multiply $2$ by $16$ .

$f'' (2) = \frac{2 \cdot 4}{32} + \frac{42 \cdot 2}{32} + \frac{12}{32}$

Step 7.2.2

Combine the numerators over the common denominator.

$f'' (2) = \frac{2 \cdot 4 + 42 \cdot 2 + 12}{32}$

Step 7.2.3

Simplify each term.

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Step 7.2.3.1

Multiply $2$ by $4$ .

$f'' (2) = \frac{8 + 42 \cdot 2 + 12}{32}$

Step 7.2.3.2

Multiply $42$ by $2$ .

$f'' (2) = \frac{8 + 84 + 12}{32}$

Step 7.2.4

Reduce the expression by cancelling the common factors.

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Step 7.2.4.1

Add $8$ and $84$ .

$f'' (2) = \frac{92 + 12}{32}$

Step 7.2.4.2

Add $92$ and $12$ .

$f'' (2) = \frac{104}{32}$

Step 7.2.4.3

Cancel the common factor of $104$ and $32$ .

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Step 7.2.4.3.1

Factor $8$ out of $104$ .

$f'' (2) = \frac{8 (13)}{32}$

Step 7.2.4.3.2

Cancel the common factors.

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Step 7.2.4.3.2.1

Factor $8$ out of $32$ .

$f'' (2) = \frac{8 \cdot 13}{8 \cdot 4}$

Step 7.2.4.3.2.2

Cancel the common factor.

$f'' (2) = \frac{8 \cdot 13}{8 \cdot 4}$

Step 7.2.4.3.2.3

Rewrite the expression.

$f'' (2) = \frac{13}{4}$

Step 7.2.5

The final answer is $\frac{13}{4}$ .

$\frac{13}{4}$

Step 7.3

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

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

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 $(- \infty, - \frac{21 + \sqrt{417}}{2})$ since $f'' (x)$ is negative

Concave up on $(- \frac{21 + \sqrt{417}}{2}, - \frac{21 - \sqrt{417}}{2})$ since $f'' (x)$ is positive

Concave down on $(- \frac{21 - \sqrt{417}}{2}, 0)$ since $f'' (x)$ is negative

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 9