Calculus Examples

$f (x) = x e^{\frac{- x^{2}}{8}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [x e^{- \frac{x^{2}}{8}}]$

Step 1.2

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 $f (x) = x$ and $g (x) = e^{- \frac{x^{2}}{8}}$ .

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

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{8}$ .

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Step 1.3.1

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

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

Step 1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 1.3.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{8}$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

Since $- \frac{1}{8}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{8}$ with respect to $x$ is $- \frac{1}{8} \frac{d}{d x} [x^{2}]$ .

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

Step 1.4.2

Combine fractions.

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Step 1.4.2.1

Combine $e^{- \frac{x^{2}}{8}}$ and $\frac{1}{8}$ .

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

Step 1.4.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{8}}}{8}$ .

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

Step 1.4.3

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

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

Step 1.4.4

Combine fractions.

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Step 1.4.4.1

Multiply $2$ by $- 1$ .

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

Step 1.4.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{8}}}{8}$ .

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

Step 1.4.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{8}})}{8}$ and $x$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Reduce the expression by cancelling the common factors.

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Step 1.8.1

Add $1$ and $1$ .

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

Step 1.8.2

Cancel the common factor of $- 2$ and $8$ .

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Step 1.8.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{8}}$ .

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

Step 1.8.2.2

Cancel the common factors.

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Step 1.8.2.2.1

Factor $2$ out of $8$ .

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

Step 1.8.2.2.2

Cancel the common factor.

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

Step 1.8.2.2.3

Rewrite the expression.

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

Step 1.8.3

Move the negative in front of the fraction.

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

Step 1.9

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

$- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} \cdot 1$

Step 1.10

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4}] + \frac{d}{d x} [e^{- \frac{x^{2}}{8}}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4}]$ .

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Step 2.2.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [x^{2} e^{- \frac{x^{2}}{8}}]$ .

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

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = e^{- \frac{x^{2}}{8}}$ .

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

Step 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 $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{8}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $- \frac{x^{2}}{8}$ .

$f'' (x) = - \frac{1}{4} \cdot (x^{2} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (- \frac{x^{2}}{8})) + e^{- \frac{x^{2}}{8}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{8}})$

Step 2.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = - \frac{1}{4} \cdot (x^{2} (e^{u_{1}} \frac{d}{d x} (- \frac{x^{2}}{8})) + e^{- \frac{x^{2}}{8}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{8}})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $- \frac{x^{2}}{8}$ .

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

Step 2.2.4

Since $- \frac{1}{8}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{8}$ with respect to $x$ is $- \frac{1}{8} \frac{d}{d x} [x^{2}]$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Multiply $2$ by $- 1$ .

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

Step 2.2.8

Combine $- 2$ and $\frac{1}{8}$ .

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

Step 2.2.9

Combine $\frac{- 2}{8}$ and $x$ .

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

Step 2.2.10

Cancel the common factor of $- 2$ and $8$ .

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Step 2.2.10.1

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

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

Step 2.2.10.2

Cancel the common factors.

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Step 2.2.10.2.1

Factor $2$ out of $8$ .

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

Step 2.2.10.2.2

Cancel the common factor.

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

Step 2.2.10.2.3

Rewrite the expression.

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

Step 2.2.11

Move the negative in front of the fraction.

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

Step 2.2.12

Combine $e^{- \frac{x^{2}}{8}}$ and $\frac{x}{4}$ .

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

Step 2.2.13

Combine $x^{2}$ and $\frac{e^{- \frac{x^{2}}{8}} x}{4}$ .

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

Step 2.2.14

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

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Step 2.2.14.1

Move $x$ .

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

Step 2.2.14.2

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

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Step 2.2.14.2.1

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

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

Step 2.2.14.2.2

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

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

Step 2.2.14.3

Add $1$ and $2$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{8}})$

Step 2.3

Evaluate $\frac{d}{d x} [e^{- \frac{x^{2}}{8}}]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{8}$ .

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Step 2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $- \frac{x^{2}}{8}$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + \frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (- \frac{x^{2}}{8})$

Step 2.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{u_{2}} \frac{d}{d x} (- \frac{x^{2}}{8})$

Step 2.3.1.3

Replace all occurrences of $u_{2}$ with $- \frac{x^{2}}{8}$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} \frac{d}{d x} (- \frac{x^{2}}{8})$

Step 2.3.2

Since $- \frac{1}{8}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{8}$ with respect to $x$ is $- \frac{1}{8} \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (- \frac{1}{8} \cdot \frac{d}{d x} x^{2})$

Step 2.3.3

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

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (- \frac{1}{8} \cdot (2 x))$

Step 2.3.4

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (- 2 (\frac{1}{8}) \cdot x)$

Step 2.3.5

Combine $- 2$ and $\frac{1}{8}$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{- 2}{8} x)$

Step 2.3.6

Combine $\frac{- 2}{8}$ and $x$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{- 2 x}{8})$

Step 2.3.7

Cancel the common factor of $- 2$ and $8$ .

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Step 2.3.7.1

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

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{2 (- x)}{8})$

Step 2.3.7.2

Cancel the common factors.

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Step 2.3.7.2.1

Factor $2$ out of $8$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{2 (- x)}{2 (4)})$

Step 2.3.7.2.2

Cancel the common factor.

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{2 (- x)}{2 \cdot 4})$

Step 2.3.7.2.3

Rewrite the expression.

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (\frac{- x}{4})$

Step 2.3.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) + e^{- \frac{x^{2}}{8}} (- \frac{x}{4})$

Step 2.3.9

Combine $e^{- \frac{x^{2}}{8}}$ and $\frac{x}{4}$ .

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = - \frac{1}{4} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4}) - \frac{1}{4} \cdot (e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{1}{4}) \cdot \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} - \frac{1}{4} \cdot (e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.2

Multiply $\frac{1}{4}$ by $1$ .

$f'' (x) = \frac{1}{4} \cdot \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4} - \frac{1}{4} \cdot (e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.3

Multiply $\frac{1}{4}$ by $\frac{x^{3} e^{- \frac{x^{2}}{8}}}{4}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{4 \cdot 4} - \frac{1}{4} \cdot (e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.4

Multiply $4$ by $4$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{1}{4} \cdot (e^{- \frac{x^{2}}{8}} (2 x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.5

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - 2 (\frac{1}{4}) \cdot (e^{- \frac{x^{2}}{8}} (x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.6

Combine $- 2$ and $\frac{1}{4}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- 2}{4} \cdot (e^{- \frac{x^{2}}{8}} (x)) - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.7

Combine $e^{- \frac{x^{2}}{8}}$ and $\frac{- 2}{4}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{e^{- \frac{x^{2}}{8}} \cdot - 2}{4} \cdot x - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.8

Combine $\frac{e^{- \frac{x^{2}}{8}} \cdot - 2}{4}$ and $x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{e^{- \frac{x^{2}}{8}} \cdot (- 2 x)}{4} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.9

Move $- 2$ to the left of $e^{- \frac{x^{2}}{8}}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- 2 \cdot (e^{- \frac{x^{2}}{8}} x)}{4} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.10

Cancel the common factor of $- 2$ and $4$ .

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Step 2.4.2.10.1

Factor $2$ out of $- 2 e^{- \frac{x^{2}}{8}} x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{2 (- e^{- \frac{x^{2}}{8}} x)}{4} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.10.2

Cancel the common factors.

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Step 2.4.2.10.2.1

Factor $2$ out of $4$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{2 (- e^{- \frac{x^{2}}{8}} x)}{2 (2)} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.10.2.2

Cancel the common factor.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{2 (- e^{- \frac{x^{2}}{8}} x)}{2 \cdot 2} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.10.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- e^{- \frac{x^{2}}{8}} x}{2} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.11

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{e^{- \frac{x^{2}}{8}} x}{2} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.12

To write $- \frac{e^{- \frac{x^{2}}{8}} x}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{e^{- \frac{x^{2}}{8}} x}{2} \cdot \frac{2}{2} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.13

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.2.13.1

Multiply $\frac{e^{- \frac{x^{2}}{8}} x}{2}$ by $\frac{2}{2}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{e^{- \frac{x^{2}}{8}} x \cdot 2}{2 \cdot 2} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.13.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{e^{- \frac{x^{2}}{8}} x \cdot 2}{4} - \frac{e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.14

Combine the numerators over the common denominator.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- e^{- \frac{x^{2}}{8}} x \cdot 2 - e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.15

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- 2 e^{- \frac{x^{2}}{8}} x - e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.16

Subtract $e^{- \frac{x^{2}}{8}} x$ from $- 2 e^{- \frac{x^{2}}{8}} x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} + \frac{- 3 e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.2.17

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{3 e^{- \frac{x^{2}}{8}} x}{4}$

Step 2.4.3

Reorder factors in $\frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{3 e^{- \frac{x^{2}}{8}} x}{4}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{8}}}{16} - \frac{3 x e^{- \frac{x^{2}}{8}}}{4}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [x e^{- \frac{x^{2}}{8}}]$

Step 4.1.2

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

Step 4.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{8}$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 4.1.3.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{8}$ .

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

Step 4.1.4

Differentiate.

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Step 4.1.4.1

Since $- \frac{1}{8}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{8}$ with respect to $x$ is $- \frac{1}{8} \frac{d}{d x} [x^{2}]$ .

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

Step 4.1.4.2

Combine fractions.

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Step 4.1.4.2.1

Combine $e^{- \frac{x^{2}}{8}}$ and $\frac{1}{8}$ .

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

Step 4.1.4.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{8}}}{8}$ .

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

Step 4.1.4.3

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

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

Step 4.1.4.4

Combine fractions.

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Step 4.1.4.4.1

Multiply $2$ by $- 1$ .

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

Step 4.1.4.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{8}}}{8}$ .

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

Step 4.1.4.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{8}})}{8}$ and $x$ .

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

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

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

Step 4.1.8

Reduce the expression by cancelling the common factors.

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Step 4.1.8.1

Add $1$ and $1$ .

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

Step 4.1.8.2

Cancel the common factor of $- 2$ and $8$ .

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Step 4.1.8.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{8}}$ .

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

Step 4.1.8.2.2

Cancel the common factors.

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Step 4.1.8.2.2.1

Factor $2$ out of $8$ .

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

Step 4.1.8.2.2.2

Cancel the common factor.

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

Step 4.1.8.2.2.3

Rewrite the expression.

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

Step 4.1.8.3

Move the negative in front of the fraction.

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

Step 4.1.9

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

$- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} \cdot 1$

Step 4.1.10

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

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

Step 4.2

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

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

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}} = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $e^{- \frac{x^{2}}{8}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4} + e^{- \frac{x^{2}}{8}}$ .

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Step 5.2.1.1

Factor $e^{- \frac{x^{2}}{8}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{8}}}{4}$ .

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

Step 5.2.1.2

Multiply by $1$ .

$e^{- \frac{x^{2}}{8}} (- \frac{x^{2}}{4}) + e^{- \frac{x^{2}}{8}} \cdot 1 = 0$

Step 5.2.1.3

Factor $e^{- \frac{x^{2}}{8}}$ out of $e^{- \frac{x^{2}}{8}} (- \frac{x^{2}}{4}) + e^{- \frac{x^{2}}{8}} \cdot 1$ .

$e^{- \frac{x^{2}}{8}} (- \frac{x^{2}}{4} + 1) = 0$

Step 5.2.2

Rewrite $1$ as $1^{2}$ .

$e^{- \frac{x^{2}}{8}} (- \frac{x^{2}}{4} + 1^{2}) = 0$

Step 5.2.3

Rewrite $\frac{x^{2}}{4}$ as ${(\frac{x}{2})}^{2}$ .

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

Step 5.2.4

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

$e^{- \frac{x^{2}}{8}} (1^{2} - {(\frac{x}{2})}^{2}) = 0$

Step 5.2.5

Factor.

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Step 5.2.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{2}$ .

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

Step 5.2.5.2

Remove unnecessary parentheses.

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

Step 5.3

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

$e^{- \frac{x^{2}}{8}} = 0$

$1 + \frac{x}{2} = 0$

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

Step 5.4

Set $e^{- \frac{x^{2}}{8}}$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $e^{- \frac{x^{2}}{8}}$ equal to $0$ .

$e^{- \frac{x^{2}}{8}} = 0$

Step 5.4.2

Solve $e^{- \frac{x^{2}}{8}} = 0$ for $x$ .

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Step 5.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- \frac{x^{2}}{8}}) = \ln (0)$

Step 5.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 5.4.2.3

There is no solution for $e^{- \frac{x^{2}}{8}} = 0$

No solution

Step 5.5

Set $1 + \frac{x}{2}$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $1 + \frac{x}{2}$ equal to $0$ .

$1 + \frac{x}{2} = 0$

Step 5.5.2

Solve $1 + \frac{x}{2} = 0$ for $x$ .

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Step 5.5.2.1

Subtract $1$ from both sides of the equation.

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

Step 5.5.2.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot - 1$

Step 5.5.2.3

Simplify both sides of the equation.

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Step 5.5.2.3.1

Simplify the left side.

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Step 5.5.2.3.1.1

Cancel the common factor of $2$ .

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Step 5.5.2.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot - 1$

Step 5.5.2.3.1.1.2

Rewrite the expression.

$x = 2 \cdot - 1$

Step 5.5.2.3.2

Simplify the right side.

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Step 5.5.2.3.2.1

Multiply $2$ by $- 1$ .

$x = - 2$

Step 5.6

Set $1 - \frac{x}{2}$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $1 - \frac{x}{2}$ equal to $0$ .

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

Step 5.6.2

Solve $1 - \frac{x}{2} = 0$ for $x$ .

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Step 5.6.2.1

Subtract $1$ from both sides of the equation.

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

Step 5.6.2.2

Multiply both sides of the equation by $- 2$ .

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

Step 5.6.2.3

Simplify both sides of the equation.

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Step 5.6.2.3.1

Simplify the left side.

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Step 5.6.2.3.1.1

Simplify $- 2 (- \frac{x}{2})$ .

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Step 5.6.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.6.2.3.1.1.1.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$- 2 \frac{- x}{2} = - 2 \cdot - 1$

Step 5.6.2.3.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 5.6.2.3.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- x}{2} = - 2 \cdot - 1$

Step 5.6.2.3.1.1.1.4

Rewrite the expression.

$- - x = - 2 \cdot - 1$

Step 5.6.2.3.1.1.2

Multiply.

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Step 5.6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 2 \cdot - 1$

Step 5.6.2.3.1.1.2.2

Multiply $x$ by $1$ .

$x = - 2 \cdot - 1$

Step 5.6.2.3.2

Simplify the right side.

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Step 5.6.2.3.2.1

Multiply $- 2$ by $- 1$ .

$x = 2$

Step 5.7

The final solution is all the values that make $e^{- \frac{x^{2}}{8}} (1 + \frac{x}{2}) (1 - \frac{x}{2}) = 0$ true.

$x = - 2, 2$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = - 2, 2$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(- 2)}^{3} e^{- \frac{{(- 2)}^{2}}{8}}}{16} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Move $e^{- \frac{{(- 2)}^{2}}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(- 2)}^{3}}{16 e^{\frac{{(- 2)}^{2}}{8}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.2

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

$\frac{{(- 2)}^{3}}{16 e^{\frac{4}{8}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.3

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

$\frac{- 8}{16 e^{\frac{4}{8}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.4

Cancel the common factor of $4$ and $8$ .

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Step 9.1.4.1

Factor $4$ out of $4$ .

$\frac{- 8}{16 e^{\frac{4 (1)}{8}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.4.2

Cancel the common factors.

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Step 9.1.4.2.1

Factor $4$ out of $8$ .

$\frac{- 8}{16 e^{\frac{4 \cdot 1}{4 \cdot 2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.4.2.2

Cancel the common factor.

$\frac{- 8}{16 e^{\frac{4 \cdot 1}{4 \cdot 2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.4.2.3

Rewrite the expression.

$\frac{- 8}{16 e^{\frac{1}{2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.5

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{16 e^{\frac{1}{2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.6

Cancel the common factors.

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Step 9.1.6.1

Factor $8$ out of $16 e^{\frac{1}{2}}$ .

$\frac{8 (- 1)}{8 (2 e^{\frac{1}{2}})} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.6.2

Cancel the common factor.

$\frac{8 \cdot - 1}{8 (2 e^{\frac{1}{2}})} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.6.3

Rewrite the expression.

$\frac{- 1}{2 e^{\frac{1}{2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.7

Move the negative in front of the fraction.

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{3 (- 2) e^{- \frac{{(- 2)}^{2}}{8}}}{4}$

Step 9.1.8

Move $e^{- \frac{{(- 2)}^{2}}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{3 (- 2)}{4 e^{\frac{{(- 2)}^{2}}{8}}}$

Step 9.1.9

Factor $2$ out of $3 (- 2)$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{2 (3 \cdot (- 1))}{4 e^{\frac{{(- 2)}^{2}}{8}}}$

Step 9.1.10

Cancel the common factors.

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Step 9.1.10.1

Factor $2$ out of $4 e^{\frac{{(- 2)}^{2}}{8}}$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{2 (3 \cdot (- 1))}{2 (2 e^{\frac{{(- 2)}^{2}}{8}})}$

Step 9.1.10.2

Cancel the common factor.

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{2 (3 \cdot (- 1))}{2 (2 e^{\frac{{(- 2)}^{2}}{8}})}$

Step 9.1.10.3

Rewrite the expression.

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{3 \cdot (- 1)}{2 e^{\frac{{(- 2)}^{2}}{8}}}$

Step 9.1.11

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

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{3 \cdot (- 1)}{2 e^{\frac{4}{8}}}$

Step 9.1.12

Multiply $3$ by $- 1$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{- 3}{2 e^{\frac{4}{8}}}$

Step 9.1.13

Cancel the common factor of $4$ and $8$ .

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Step 9.1.13.1

Factor $4$ out of $4$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{- 3}{2 e^{\frac{4 (1)}{8}}}$

Step 9.1.13.2

Cancel the common factors.

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Step 9.1.13.2.1

Factor $4$ out of $8$ .

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{- 3}{2 e^{\frac{4 \cdot 1}{4 \cdot 2}}}$

Step 9.1.13.2.2

Cancel the common factor.

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{- 3}{2 e^{\frac{4 \cdot 1}{4 \cdot 2}}}$

Step 9.1.13.2.3

Rewrite the expression.

$- \frac{1}{2 e^{\frac{1}{2}}} - \frac{- 3}{2 e^{\frac{1}{2}}}$

Step 9.1.14

Move the negative in front of the fraction.

$- \frac{1}{2 e^{\frac{1}{2}}} - - \frac{3}{2 e^{\frac{1}{2}}}$

Step 9.1.15

Multiply $- - \frac{3}{2 e^{\frac{1}{2}}}$ .

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Step 9.1.15.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{2 e^{\frac{1}{2}}} + 1 \frac{3}{2 e^{\frac{1}{2}}}$

Step 9.1.15.2

Multiply $\frac{3}{2 e^{\frac{1}{2}}}$ by $1$ .

$- \frac{1}{2 e^{\frac{1}{2}}} + \frac{3}{2 e^{\frac{1}{2}}}$

Step 9.2

Simplify terms.

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Step 9.2.1

Combine the numerators over the common denominator.

$\frac{- 1 + 3}{2 e^{\frac{1}{2}}}$

Step 9.2.2

Add $- 1$ and $3$ .

$\frac{2}{2 e^{\frac{1}{2}}}$

Step 9.2.3

Cancel the common factor.

$\frac{2}{2 e^{\frac{1}{2}}}$

Step 9.2.4

Rewrite the expression.

$\frac{1}{e^{\frac{1}{2}}}$

Step 10

$x = - 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 2$ is a local minimum

Step 11

Find the y-value when $x = - 2$ .

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Step 11.1

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

$f (- 2) = (- 2) \cdot e^{\frac{- {(- 2)}^{2}}{8}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

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

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

Step 11.2.1.2

Multiply $- 1$ by $4$ .

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

Step 11.2.2

Cancel the common factor of $- 4$ and $8$ .

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Step 11.2.2.1

Factor $4$ out of $- 4$ .

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

Step 11.2.2.2

Cancel the common factors.

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Step 11.2.2.2.1

Factor $4$ out of $8$ .

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

Step 11.2.2.2.2

Cancel the common factor.

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

Step 11.2.2.2.3

Rewrite the expression.

$f (- 2) = - 2 \cdot e^{\frac{- 1}{2}}$

Step 11.2.3

Move the negative in front of the fraction.

$f (- 2) = - 2 \cdot e^{- \frac{1}{2}}$

Step 11.2.4

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

$f (- 2) = - 2 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 11.2.5

Combine $- 2$ and $\frac{1}{e^{\frac{1}{2}}}$ .

$f (- 2) = \frac{- 2}{e^{\frac{1}{2}}}$

Step 11.2.6

Move the negative in front of the fraction.

$f (- 2) = - \frac{2}{e^{\frac{1}{2}}}$

Step 11.2.7

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

$y = - \frac{2}{e^{\frac{1}{2}}}$

Step 12

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(2)}^{3} e^{- \frac{{(2)}^{2}}{8}}}{16} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Move $e^{- \frac{{(2)}^{2}}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(2)}^{3}}{16 e^{\frac{2^{2}}{8}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.2

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

$\frac{2^{3}}{16 e^{\frac{4}{8}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.3

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

$\frac{8}{16 e^{\frac{4}{8}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.4

Cancel the common factor of $4$ and $8$ .

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Step 13.1.4.1

Factor $4$ out of $4$ .

$\frac{8}{16 e^{\frac{4 (1)}{8}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.4.2

Cancel the common factors.

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Step 13.1.4.2.1

Factor $4$ out of $8$ .

$\frac{8}{16 e^{\frac{4 \cdot 1}{4 \cdot 2}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.4.2.2

Cancel the common factor.

$\frac{8}{16 e^{\frac{4 \cdot 1}{4 \cdot 2}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.4.2.3

Rewrite the expression.

$\frac{8}{16 e^{\frac{1}{2}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.5

Factor $8$ out of $8$ .

$\frac{8 (1)}{16 e^{\frac{1}{2}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.6

Cancel the common factors.

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Step 13.1.6.1

Factor $8$ out of $16 e^{\frac{1}{2}}$ .

$\frac{8 (1)}{8 (2 e^{\frac{1}{2}})} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.6.2

Cancel the common factor.

$\frac{8 \cdot 1}{8 (2 e^{\frac{1}{2}})} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.6.3

Rewrite the expression.

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3 (2) e^{- \frac{{(2)}^{2}}{8}}}{4}$

Step 13.1.7

Move $e^{- \frac{{(2)}^{2}}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3 (2)}{4 e^{\frac{2^{2}}{8}}}$

Step 13.1.8

Factor $2$ out of $3 (2)$ .

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{2 \cdot 3}{4 e^{\frac{2^{2}}{8}}}$

Step 13.1.9

Cancel the common factors.

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Step 13.1.9.1

Factor $2$ out of $4 e^{\frac{2^{2}}{8}}$ .

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{2 \cdot 3}{2 (2 e^{\frac{2^{2}}{8}})}$

Step 13.1.9.2

Cancel the common factor.

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{2 \cdot 3}{2 (2 e^{\frac{2^{2}}{8}})}$

Step 13.1.9.3

Rewrite the expression.

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{2^{2}}{8}}}$

Step 13.1.10

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

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{4}{8}}}$

Step 13.1.11

Cancel the common factor of $4$ and $8$ .

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Step 13.1.11.1

Factor $4$ out of $4$ .

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{4 (1)}{8}}}$

Step 13.1.11.2

Cancel the common factors.

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Step 13.1.11.2.1

Factor $4$ out of $8$ .

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{4 \cdot 1}{4 \cdot 2}}}$

Step 13.1.11.2.2

Cancel the common factor.

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{4 \cdot 1}{4 \cdot 2}}}$

Step 13.1.11.2.3

Rewrite the expression.

$\frac{1}{2 e^{\frac{1}{2}}} - \frac{3}{2 e^{\frac{1}{2}}}$

Step 13.2

Simplify terms.

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Step 13.2.1

Combine the numerators over the common denominator.

$\frac{1 - 3}{2 e^{\frac{1}{2}}}$

Step 13.2.2

Subtract $3$ from $1$ .

$\frac{- 2}{2 e^{\frac{1}{2}}}$

Step 13.2.3

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2 e^{\frac{1}{2}}}$

Step 13.3

Cancel the common factors.

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Step 13.3.1

Factor $2$ out of $2 e^{\frac{1}{2}}$ .

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

Step 13.3.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 e^{\frac{1}{2}}}$

Step 13.3.3

Rewrite the expression.

$\frac{- 1}{e^{\frac{1}{2}}}$

Step 13.4

Move the negative in front of the fraction.

$- \frac{1}{e^{\frac{1}{2}}}$

Step 14

$x = 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 2$ is a local maximum

Step 15

Find the y-value when $x = 2$ .

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Step 15.1

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

$f (2) = (2) \cdot e^{\frac{- {(2)}^{2}}{8}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify the expression.

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Step 15.2.1.1

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

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

Step 15.2.1.2

Multiply $- 1$ by $4$ .

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

Step 15.2.2

Cancel the common factor of $- 4$ and $8$ .

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Step 15.2.2.1

Factor $4$ out of $- 4$ .

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

Step 15.2.2.2

Cancel the common factors.

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Step 15.2.2.2.1

Factor $4$ out of $8$ .

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

Step 15.2.2.2.2

Cancel the common factor.

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

Step 15.2.2.2.3

Rewrite the expression.

$f (2) = 2 \cdot e^{\frac{- 1}{2}}$

Step 15.2.3

Move the negative in front of the fraction.

$f (2) = 2 \cdot e^{- \frac{1}{2}}$

Step 15.2.4

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

$f (2) = 2 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 15.2.5

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

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

Step 15.2.6

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

$y = \frac{2}{e^{\frac{1}{2}}}$

Step 16

These are the local extrema for $f (x) = x e^{\frac{- x^{2}}{8}}$ .

$(- 2, - \frac{2}{e^{\frac{1}{2}}})$ is a local minima

$(2, \frac{2}{e^{\frac{1}{2}}})$ is a local maxima

Step 17