Calculus Examples

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

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

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

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

Step 1.1.1.2

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 $g (x) = 1 + x^{2}$ .

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

$f' (x) = - {(1 + x^{2})}^{- 2} (2 x)$

Step 1.1.1.3.5

Multiply $2$ by $- 1$ .

$f' (x) = - 2 {(1 + x^{2})}^{- 2} x$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

Combine terms.

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Step 1.1.1.4.2.1

Combine $- 2$ and $\frac{1}{{(1 + x^{2})}^{2}}$ .

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

Step 1.1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.1.4.2.3

Combine $x$ and $\frac{2}{{(1 + x^{2})}^{2}}$ .

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

Step 1.1.1.4.2.4

Move $2$ to the left of $x$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Differentiate using the Power Rule.

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Step 1.1.2.3.1

Multiply the exponents in ${({(1 + x^{2})}^{2})}^{2}$ .

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Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

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

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

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 $f (x) = x^{2}$ and $g (x) = 1 + x^{2}$ .

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Step 1.1.2.4.1

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

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

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 $n = 2$ .

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

Step 1.1.2.4.3

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

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

Step 1.1.2.5

Simplify with factoring out.

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Step 1.1.2.5.1

Multiply $2$ by $- 1$ .

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

Step 1.1.2.5.2

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{2} - 2 x ((1 + x^{2}) \frac{d}{d x} [1 + x^{2}])$ .

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Step 1.1.2.5.2.1

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{2}$ .

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

Step 1.1.2.5.2.2

Factor $1 + x^{2}$ out of $- 2 x ((1 + x^{2}) \frac{d}{d x} [1 + x^{2}])$ .

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

Step 1.1.2.5.2.3

Factor $1 + x^{2}$ out of $(1 + x^{2}) (1 + x^{2}) + (1 + x^{2}) (- 2 x (\frac{d}{d x} [1 + x^{2}]))$ .

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

Step 1.1.2.6

Cancel the common factors.

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Step 1.1.2.6.1

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{4}$ .

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

Step 1.1.2.6.2

Cancel the common factor.

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

Step 1.1.2.6.3

Rewrite the expression.

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 1.1.2.10

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

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

Step 1.1.2.11

Multiply $2$ by $- 2$ .

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

Step 1.1.2.12

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

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

Step 1.1.2.13

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

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

Step 1.1.2.14

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

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

Step 1.1.2.15

Add $1$ and $1$ .

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

Step 1.1.2.16

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

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

Step 1.1.2.17

Combine $- 2$ and $\frac{1 - 3 x^{2}}{{(1 + x^{2})}^{3}}$ .

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

Step 1.1.2.18

Move the negative in front of the fraction.

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

Step 1.1.2.19

Simplify.

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Step 1.1.2.19.1

Apply the distributive property.

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

Step 1.1.2.19.2

Simplify each term.

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Step 1.1.2.19.2.1

Multiply $2$ by $1$ .

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

Step 1.1.2.19.2.2

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{2 - 6 x^{2}}{{(1 + x^{2})}^{3}}$

Step 1.1.2.19.3

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

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Step 1.1.2.19.3.1

Factor $2$ out of $2$ .

$f'' (x) = - \frac{2 (1) - 6 x^{2}}{{(1 + x^{2})}^{3}}$

Step 1.1.2.19.3.2

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

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

Step 1.1.2.19.3.3

Factor $2$ out of $2 (1) + 2 (- 3 x^{2})$ .

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2 (1 - 3 x^{2})}{{(1 + x^{2})}^{3}}$ .

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

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2 (1 - 3 x^{2})}{{(1 + x^{2})}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$2 (1 - 3 x^{2}) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $2 (1 - 3 x^{2}) = 0$ by $2$ and simplify.

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Step 1.2.3.1.1

Divide each term in $2 (1 - 3 x^{2}) = 0$ by $2$ .

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

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 $2$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

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

Step 1.2.3.1.2.1.2

Divide $1 - 3 x^{2}$ by $1$ .

$1 - 3 x^{2} = \frac{0}{2}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $2$ .

$1 - 3 x^{2} = 0$

Step 1.2.3.2

Subtract $1$ from both sides of the equation.

$- 3 x^{2} = - 1$

Step 1.2.3.3

Divide each term in $- 3 x^{2} = - 1$ by $- 3$ and simplify.

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Step 1.2.3.3.1

Divide each term in $- 3 x^{2} = - 1$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{- 1}{- 3}$

Step 1.2.3.3.2

Simplify the left side.

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Step 1.2.3.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- 1}{- 3}$

Step 1.2.3.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{- 3}$

Step 1.2.3.3.3

Simplify the right side.

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Step 1.2.3.3.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{3}$

Step 1.2.3.4

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 1.2.3.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 1.2.3.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 1.2.3.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 1.2.3.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.2.3.5.4

Combine and simplify the denominator.

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Step 1.2.3.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.3.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.3.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.3.5.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.3.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.3.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.3.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.2.3.5.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 1.2.3.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 1.2.3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{3}}{3}$

Step 1.2.3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{3}}{3}$

Step 1.2.3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 2

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

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Step 2.1

Set the denominator in $\frac{1}{1 + x^{2}}$ equal to $0$ to find where the expression is undefined.

$1 + x^{2} = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.2.2

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

$x = \pm \sqrt{- 1}$

Step 2.2.3

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

$x = \pm i$

Step 2.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 2.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 2.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 2.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

$(- \infty, - \frac{\sqrt{3}}{3}) \cup (- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}) \cup (\frac{\sqrt{3}}{3}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 3$ by ${(- 3)}^{2}$ by adding the exponents.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Raise $- 3$ to the power of $1$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

Add $1$ and $2$ .

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

Step 4.2.2

Simplify the numerator.

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Step 4.2.2.1

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

$f'' (- 3) = - \frac{2 (1 - 27)}{{(1 + {(- 3)}^{2})}^{3}}$

Step 4.2.2.2

Subtract $27$ from $1$ .

$f'' (- 3) = - \frac{2 \cdot - 26}{{(1 + {(- 3)}^{2})}^{3}}$

Step 4.2.3

Simplify the denominator.

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Step 4.2.3.1

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

$f'' (- 3) = - \frac{2 \cdot - 26}{{(1 + 9)}^{3}}$

Step 4.2.3.2

Add $1$ and $9$ .

$f'' (- 3) = - \frac{2 \cdot - 26}{10^{3}}$

Step 4.2.3.3

Raise $10$ to the power of $3$ .

$f'' (- 3) = - \frac{2 \cdot - 26}{1000}$

Step 4.2.4

Reduce the expression by cancelling the common factors.

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Step 4.2.4.1

Multiply $2$ by $- 26$ .

$f'' (- 3) = - \frac{- 52}{1000}$

Step 4.2.4.2

Cancel the common factor of $- 52$ and $1000$ .

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Step 4.2.4.2.1

Factor $4$ out of $- 52$ .

$f'' (- 3) = - \frac{4 (- 13)}{1000}$

Step 4.2.4.2.2

Cancel the common factors.

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Step 4.2.4.2.2.1

Factor $4$ out of $1000$ .

$f'' (- 3) = - \frac{4 \cdot - 13}{4 \cdot 250}$

Step 4.2.4.2.2.2

Cancel the common factor.

$f'' (- 3) = - \frac{4 \cdot - 13}{4 \cdot 250}$

Step 4.2.4.2.2.3

Rewrite the expression.

$f'' (- 3) = - \frac{- 13}{250}$

Step 4.2.4.3

Move the negative in front of the fraction.

$f'' (- 3) = \frac{13}{250}$

Step 4.2.5

The final answer is $\frac{13}{250}$ .

$\frac{13}{250}$

Step 4.3

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

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

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

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

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

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

Step 5.2.1.2

Multiply $- 3$ by $0$ .

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

Step 5.2.1.3

Add $1$ and $0$ .

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

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Add $1$ and $0$ .

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

Step 5.2.2.3

One to any power is one.

$f'' (0) = - \frac{2 \cdot 1}{1}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $1$ .

$f'' (0) = - \frac{2}{1}$

Step 5.2.3.2

Divide $2$ by $1$ .

$f'' (0) = - 1 \cdot 2$

Step 5.2.3.3

Multiply $- 1$ by $2$ .

$f'' (0) = - 2$

Step 5.2.4

The final answer is $- 2$ .

$- 2$

Step 5.3

The graph is concave down on the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ because $f'' (0)$ is negative.

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(\frac{\sqrt{3}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

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

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

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

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

Step 6.2.1.2

Multiply $- 3$ by $9$ .

$f'' (3) = - \frac{2 (1 - 27)}{{(1 + 3^{2})}^{3}}$

Step 6.2.1.3

Subtract $27$ from $1$ .

$f'' (3) = - \frac{2 \cdot - 26}{{(1 + 3^{2})}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f'' (3) = - \frac{2 \cdot - 26}{{(1 + 9)}^{3}}$

Step 6.2.2.2

Add $1$ and $9$ .

$f'' (3) = - \frac{2 \cdot - 26}{10^{3}}$

Step 6.2.2.3

Raise $10$ to the power of $3$ .

$f'' (3) = - \frac{2 \cdot - 26}{1000}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $2$ by $- 26$ .

$f'' (3) = - \frac{- 52}{1000}$

Step 6.2.3.2

Cancel the common factor of $- 52$ and $1000$ .

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Step 6.2.3.2.1

Factor $4$ out of $- 52$ .

$f'' (3) = - \frac{4 (- 13)}{1000}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $4$ out of $1000$ .

$f'' (3) = - \frac{4 \cdot - 13}{4 \cdot 250}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (3) = - \frac{4 \cdot - 13}{4 \cdot 250}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (3) = - \frac{- 13}{250}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f'' (3) = \frac{13}{250}$

Step 6.2.4

The final answer is $\frac{13}{250}$ .

$\frac{13}{250}$

Step 6.3

The graph is concave up on the interval $(\frac{\sqrt{3}}{3}, \infty)$ because $f'' (3)$ is positive.

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

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 $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 8