Calculus Examples

$x e^{- \frac{x^{2}}{162}}$

Step 1

Write $x e^{- \frac{x^{2}}{162}}$ as a function.

$f (x) = x e^{- \frac{x^{2}}{162}}$

Step 2

Find the first derivative of the function.

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Step 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 $f (x) = x$ and $g (x) = e^{- \frac{x^{2}}{162}}$ .

$x \frac{d}{d x} [e^{- \frac{x^{2}}{162}}] + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{162}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{162}$ .

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.2.2

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

$x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{162}$ .

$x (e^{- \frac{x^{2}}{162}} \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3

Differentiate.

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Step 2.3.1

Since $- \frac{1}{162}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{162}$ with respect to $x$ is $- \frac{1}{162} \frac{d}{d x} [x^{2}]$ .

$x (e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Combine $e^{- \frac{x^{2}}{162}}$ and $\frac{1}{162}$ .

$x (- \frac{e^{- \frac{x^{2}}{162}}}{162} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{162}}}{162}$ .

$- \frac{x e^{- \frac{x^{2}}{162}}}{162} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.3

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

$- \frac{x e^{- \frac{x^{2}}{162}}}{162} (2 x) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{x e^{- \frac{x^{2}}{162}}}{162} x + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{162}}}{162}$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{162}})}{162} x + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.3.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{162}})}{162}$ and $x$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{162}}) x}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.4

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

$\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.5

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

$\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.6

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

$\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

Add $1$ and $1$ .

$\frac{- 2 (x^{2} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7.2

Cancel the common factor of $- 2$ and $162$ .

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Step 2.7.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{162}}$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7.2.2

Cancel the common factors.

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Step 2.7.2.2.1

Factor $2$ out of $162$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{2 (81)} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7.2.2.2

Cancel the common factor.

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{2 \cdot 81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7.2.2.3

Rewrite the expression.

$\frac{- x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.7.3

Move the negative in front of the fraction.

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 2.8

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

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \cdot 1$

Step 2.9

Multiply $e^{- \frac{x^{2}}{162}}$ by $1$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81}] + \frac{d}{d x} [e^{- \frac{x^{2}}{162}}]$ .

$f'' (x) = \frac{d}{d x} (- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81}) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81}]$ .

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Step 3.2.1

Since $- \frac{1}{81}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81}$ with respect to $x$ is $- \frac{1}{81} \frac{d}{d x} [x^{2} e^{- \frac{x^{2}}{162}}]$ .

$f'' (x) = - \frac{1}{81} \cdot \frac{d}{d x} (x^{2} e^{- \frac{x^{2}}{162}}) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.2

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 $g (x) = e^{- \frac{x^{2}}{162}}$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} \frac{d}{d x} (e^{- \frac{x^{2}}{162}}) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.3

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 $g (x) = - \frac{x^{2}}{162}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $- \frac{x^{2}}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (- \frac{x^{2}}{162})) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.3.2

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

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{u_{1}} \frac{d}{d x} (- \frac{x^{2}}{162})) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $- \frac{x^{2}}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} \frac{d}{d x} (- \frac{x^{2}}{162})) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.4

Since $- \frac{1}{162}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{162}$ with respect to $x$ is $- \frac{1}{162} \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \cdot \frac{d}{d x} x^{2})) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.5

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

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \cdot (2 x))) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.6

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

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \cdot (2 x))) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.7

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (- 2 (\frac{1}{162}) \cdot x)) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.8

Combine $- 2$ and $\frac{1}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{- 2}{162} x)) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.9

Combine $\frac{- 2}{162}$ and $x$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{- 2 x}{162})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.10

Cancel the common factor of $- 2$ and $162$ .

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Step 3.2.10.1

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

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{162})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.10.2

Cancel the common factors.

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Step 3.2.10.2.1

Factor $2$ out of $162$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{2 (81)})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.10.2.2

Cancel the common factor.

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{2 \cdot 81})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.10.2.3

Rewrite the expression.

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (\frac{- x}{81})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (e^{- \frac{x^{2}}{162}} (- \frac{x}{81})) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.12

Combine $e^{- \frac{x^{2}}{162}}$ and $\frac{x}{81}$ .

$f'' (x) = - \frac{1}{81} \cdot (x^{2} (- \frac{e^{- \frac{x^{2}}{162}} x}{81}) + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.13

Combine $x^{2}$ and $\frac{e^{- \frac{x^{2}}{162}} x}{81}$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{2} (e^{- \frac{x^{2}}{162}} x)}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.14

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

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Step 3.2.14.1

Move $x$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x \cdot x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.14.2

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

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Step 3.2.14.2.1

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

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x \cdot x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.14.2.2

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

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{1 + 2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.2.14.3

Add $1$ and $2$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{162}})$

Step 3.3

Evaluate $\frac{d}{d x} [e^{- \frac{x^{2}}{162}}]$ .

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Step 3.3.1

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 $g (x) = - \frac{x^{2}}{162}$ .

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Step 3.3.1.1

To apply the Chain Rule, set $u_{2}$ as $- \frac{x^{2}}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + \frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (- \frac{x^{2}}{162})$

Step 3.3.1.2

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

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{u_{2}} \frac{d}{d x} (- \frac{x^{2}}{162})$

Step 3.3.1.3

Replace all occurrences of $u_{2}$ with $- \frac{x^{2}}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} (- \frac{x^{2}}{162})$

Step 3.3.2

Since $- \frac{1}{162}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{162}$ with respect to $x$ is $- \frac{1}{162} \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \cdot \frac{d}{d x} x^{2})$

Step 3.3.3

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

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \cdot (2 x))$

Step 3.3.4

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (- 2 (\frac{1}{162}) \cdot x)$

Step 3.3.5

Combine $- 2$ and $\frac{1}{162}$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{- 2}{162} x)$

Step 3.3.6

Combine $\frac{- 2}{162}$ and $x$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{- 2 x}{162})$

Step 3.3.7

Cancel the common factor of $- 2$ and $162$ .

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Step 3.3.7.1

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

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{162})$

Step 3.3.7.2

Cancel the common factors.

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Step 3.3.7.2.1

Factor $2$ out of $162$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{2 (81)})$

Step 3.3.7.2.2

Cancel the common factor.

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{2 (- x)}{2 \cdot 81})$

Step 3.3.7.2.3

Rewrite the expression.

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (\frac{- x}{81})$

Step 3.3.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) + e^{- \frac{x^{2}}{162}} (- \frac{x}{81})$

Step 3.3.9

Combine $e^{- \frac{x^{2}}{162}}$ and $\frac{x}{81}$ .

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$f'' (x) = - \frac{1}{81} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81}) - \frac{1}{81} \cdot (e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{1}{81}) \cdot \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} - \frac{1}{81} \cdot (e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.2

Multiply $\frac{1}{81}$ by $1$ .

$f'' (x) = \frac{1}{81} \cdot \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81} - \frac{1}{81} \cdot (e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.3

Multiply $\frac{1}{81}$ by $\frac{x^{3} e^{- \frac{x^{2}}{162}}}{81}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{81 \cdot 81} - \frac{1}{81} \cdot (e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.4

Multiply $81$ by $81$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - \frac{1}{81} \cdot (e^{- \frac{x^{2}}{162}} (2 x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.5

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - 2 (\frac{1}{81}) \cdot (e^{- \frac{x^{2}}{162}} (x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.6

Combine $- 2$ and $\frac{1}{81}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{- 2}{81} \cdot (e^{- \frac{x^{2}}{162}} (x)) - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.7

Combine $e^{- \frac{x^{2}}{162}}$ and $\frac{- 2}{81}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{e^{- \frac{x^{2}}{162}} \cdot - 2}{81} \cdot x - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.8

Combine $\frac{e^{- \frac{x^{2}}{162}} \cdot - 2}{81}$ and $x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{e^{- \frac{x^{2}}{162}} \cdot (- 2 x)}{81} - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.9

Move $- 2$ to the left of $e^{- \frac{x^{2}}{162}}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{- 2 \cdot (e^{- \frac{x^{2}}{162}} x)}{81} - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.10

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - \frac{2 e^{- \frac{x^{2}}{162}} x}{81} - \frac{e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.11

Combine the numerators over the common denominator.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{- 2 e^{- \frac{x^{2}}{162}} x - e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.12

Subtract $e^{- \frac{x^{2}}{162}} x$ from $- 2 e^{- \frac{x^{2}}{162}} x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{- 3 e^{- \frac{x^{2}}{162}} x}{81}$

Step 3.4.2.13

Cancel the common factor of $- 3$ and $81$ .

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Step 3.4.2.13.1

Factor $3$ out of $- 3 e^{- \frac{x^{2}}{162}} x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{3 (- e^{- \frac{x^{2}}{162}} x)}{81}$

Step 3.4.2.13.2

Cancel the common factors.

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Step 3.4.2.13.2.1

Factor $3$ out of $81$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{3 (- e^{- \frac{x^{2}}{162}} x)}{3 (27)}$

Step 3.4.2.13.2.2

Cancel the common factor.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{3 (- e^{- \frac{x^{2}}{162}} x)}{3 \cdot 27}$

Step 3.4.2.13.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} + \frac{- e^{- \frac{x^{2}}{162}} x}{27}$

Step 3.4.2.14

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - \frac{e^{- \frac{x^{2}}{162}} x}{27}$

Step 3.4.3

Reorder factors in $\frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - \frac{e^{- \frac{x^{2}}{162}} x}{27}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{162}}}{6561} - \frac{x e^{- \frac{x^{2}}{162}}}{27}$

Step 4

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

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$x \frac{d}{d x} [e^{- \frac{x^{2}}{162}}] + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x^{2}}{162}$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{162}$ .

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.2.2

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

$x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.2.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{162}$ .

$x (e^{- \frac{x^{2}}{162}} \frac{d}{d x} [- \frac{x^{2}}{162}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

Since $- \frac{1}{162}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{162}$ with respect to $x$ is $- \frac{1}{162} \frac{d}{d x} [x^{2}]$ .

$x (e^{- \frac{x^{2}}{162}} (- \frac{1}{162} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.2

Combine fractions.

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Step 5.1.3.2.1

Combine $e^{- \frac{x^{2}}{162}}$ and $\frac{1}{162}$ .

$x (- \frac{e^{- \frac{x^{2}}{162}}}{162} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{162}}}{162}$ .

$- \frac{x e^{- \frac{x^{2}}{162}}}{162} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.3

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

$- \frac{x e^{- \frac{x^{2}}{162}}}{162} (2 x) + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.4

Combine fractions.

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Step 5.1.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{x e^{- \frac{x^{2}}{162}}}{162} x + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{162}}}{162}$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{162}})}{162} x + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.3.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{162}})}{162}$ and $x$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{162}}) x}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.4

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

$\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.5

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

$\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.6

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

$\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7

Reduce the expression by cancelling the common factors.

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Step 5.1.7.1

Add $1$ and $1$ .

$\frac{- 2 (x^{2} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7.2

Cancel the common factor of $- 2$ and $162$ .

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Step 5.1.7.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{162}}$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{162} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7.2.2

Cancel the common factors.

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Step 5.1.7.2.2.1

Factor $2$ out of $162$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{2 (81)} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7.2.2.2

Cancel the common factor.

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{162}})}{2 \cdot 81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7.2.2.3

Rewrite the expression.

$\frac{- x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.7.3

Move the negative in front of the fraction.

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \frac{d}{d x} [x]$

Step 5.1.8

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

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} \cdot 1$

Step 5.1.9

Multiply $e^{- \frac{x^{2}}{162}}$ by $1$ .

$f' (x) = - \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}} = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Factor $e^{- \frac{x^{2}}{162}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81} + e^{- \frac{x^{2}}{162}}$ .

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Step 6.2.1.1

Factor $e^{- \frac{x^{2}}{162}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{162}}}{81}$ .

$e^{- \frac{x^{2}}{162}} (- \frac{x^{2}}{81}) + e^{- \frac{x^{2}}{162}} = 0$

Step 6.2.1.2

Multiply by $1$ .

$e^{- \frac{x^{2}}{162}} (- \frac{x^{2}}{81}) + e^{- \frac{x^{2}}{162}} \cdot 1 = 0$

Step 6.2.1.3

Factor $e^{- \frac{x^{2}}{162}}$ out of $e^{- \frac{x^{2}}{162}} (- \frac{x^{2}}{81}) + e^{- \frac{x^{2}}{162}} \cdot 1$ .

$e^{- \frac{x^{2}}{162}} (- \frac{x^{2}}{81} + 1) = 0$

Step 6.2.2

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

$e^{- \frac{x^{2}}{162}} (- \frac{x^{2}}{81} + 1^{2}) = 0$

Step 6.2.3

Rewrite $\frac{x^{2}}{81}$ as ${(\frac{x}{9})}^{2}$ .

$e^{- \frac{x^{2}}{162}} (- {(\frac{x}{9})}^{2} + 1^{2}) = 0$

Step 6.2.4

Reorder $- {(\frac{x}{9})}^{2}$ and $1^{2}$ .

$e^{- \frac{x^{2}}{162}} (1^{2} - {(\frac{x}{9})}^{2}) = 0$

Step 6.2.5

Factor.

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Step 6.2.5.1

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 $b = \frac{x}{9}$ .

$e^{- \frac{x^{2}}{162}} ((1 + \frac{x}{9}) (1 - \frac{x}{9})) = 0$

Step 6.2.5.2

Remove unnecessary parentheses.

$e^{- \frac{x^{2}}{162}} (1 + \frac{x}{9}) (1 - \frac{x}{9}) = 0$

Step 6.3

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

$e^{- \frac{x^{2}}{162}} = 0$

$1 + \frac{x}{9} = 0$

$1 - \frac{x}{9} = 0$

Step 6.4

Set $e^{- \frac{x^{2}}{162}}$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $e^{- \frac{x^{2}}{162}}$ equal to $0$ .

$e^{- \frac{x^{2}}{162}} = 0$

Step 6.4.2

Solve $e^{- \frac{x^{2}}{162}} = 0$ for $x$ .

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Step 6.4.2.1

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

$\ln (e^{- \frac{x^{2}}{162}}) = \ln (0)$

Step 6.4.2.2

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

Undefined

Step 6.4.2.3

There is no solution for $e^{- \frac{x^{2}}{162}} = 0$

No solution

Step 6.5

Set $1 + \frac{x}{9}$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $1 + \frac{x}{9}$ equal to $0$ .

$1 + \frac{x}{9} = 0$

Step 6.5.2

Solve $1 + \frac{x}{9} = 0$ for $x$ .

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Step 6.5.2.1

Subtract $1$ from both sides of the equation.

$\frac{x}{9} = - 1$

Step 6.5.2.2

Multiply both sides of the equation by $9$ .

$9 \frac{x}{9} = 9 \cdot - 1$

Step 6.5.2.3

Simplify both sides of the equation.

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Step 6.5.2.3.1

Simplify the left side.

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Step 6.5.2.3.1.1

Cancel the common factor of $9$ .

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Step 6.5.2.3.1.1.1

Cancel the common factor.

$9 \frac{x}{9} = 9 \cdot - 1$

Step 6.5.2.3.1.1.2

Rewrite the expression.

$x = 9 \cdot - 1$

Step 6.5.2.3.2

Simplify the right side.

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Step 6.5.2.3.2.1

Multiply $9$ by $- 1$ .

$x = - 9$

Step 6.6

Set $1 - \frac{x}{9}$ equal to $0$ and solve for $x$ .

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Step 6.6.1

Set $1 - \frac{x}{9}$ equal to $0$ .

$1 - \frac{x}{9} = 0$

Step 6.6.2

Solve $1 - \frac{x}{9} = 0$ for $x$ .

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Step 6.6.2.1

Subtract $1$ from both sides of the equation.

$- \frac{x}{9} = - 1$

Step 6.6.2.2

Multiply both sides of the equation by $- 9$ .

$- 9 (- \frac{x}{9}) = - 9 \cdot - 1$

Step 6.6.2.3

Simplify both sides of the equation.

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Step 6.6.2.3.1

Simplify the left side.

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Step 6.6.2.3.1.1

Simplify $- 9 (- \frac{x}{9})$ .

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Step 6.6.2.3.1.1.1

Cancel the common factor of $9$ .

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Step 6.6.2.3.1.1.1.1

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$- 9 \frac{- x}{9} = - 9 \cdot - 1$

Step 6.6.2.3.1.1.1.2

Factor $9$ out of $- 9$ .

$9 (- 1) \frac{- x}{9} = - 9 \cdot - 1$

Step 6.6.2.3.1.1.1.3

Cancel the common factor.

$9 \cdot - 1 \frac{- x}{9} = - 9 \cdot - 1$

Step 6.6.2.3.1.1.1.4

Rewrite the expression.

$- - x = - 9 \cdot - 1$

Step 6.6.2.3.1.1.2

Multiply.

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Step 6.6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 9 \cdot - 1$

Step 6.6.2.3.1.1.2.2

Multiply $x$ by $1$ .

$x = - 9 \cdot - 1$

Step 6.6.2.3.2

Simplify the right side.

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Step 6.6.2.3.2.1

Multiply $- 9$ by $- 1$ .

$x = 9$

Step 6.7

The final solution is all the values that make $e^{- \frac{x^{2}}{162}} (1 + \frac{x}{9}) (1 - \frac{x}{9}) = 0$ true.

$x = - 9, 9$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 8

Critical points to evaluate.

$x = - 9, 9$

Step 9

Evaluate the second derivative at $x = - 9$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(- 9)}^{3} e^{- \frac{{(- 9)}^{2}}{162}}}{6561} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Move $e^{- \frac{{(- 9)}^{2}}{162}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(- 9)}^{3}}{6561 e^{\frac{{(- 9)}^{2}}{162}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.2

Raise $- 9$ to the power of $2$ .

$\frac{{(- 9)}^{3}}{6561 e^{\frac{81}{162}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.3

Raise $- 9$ to the power of $3$ .

$\frac{- 729}{6561 e^{\frac{81}{162}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.4

Cancel the common factor of $81$ and $162$ .

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Step 10.1.4.1

Factor $81$ out of $81$ .

$\frac{- 729}{6561 e^{\frac{81 (1)}{162}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.4.2

Cancel the common factors.

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Step 10.1.4.2.1

Factor $81$ out of $162$ .

$\frac{- 729}{6561 e^{\frac{81 \cdot 1}{81 \cdot 2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.4.2.2

Cancel the common factor.

$\frac{- 729}{6561 e^{\frac{81 \cdot 1}{81 \cdot 2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.4.2.3

Rewrite the expression.

$\frac{- 729}{6561 e^{\frac{1}{2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.5

Factor $729$ out of $- 729$ .

$\frac{729 (- 1)}{6561 e^{\frac{1}{2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.6

Cancel the common factors.

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Step 10.1.6.1

Factor $729$ out of $6561 e^{\frac{1}{2}}$ .

$\frac{729 (- 1)}{729 (9 e^{\frac{1}{2}})} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.6.2

Cancel the common factor.

$\frac{729 \cdot - 1}{729 (9 e^{\frac{1}{2}})} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.6.3

Rewrite the expression.

$\frac{- 1}{9 e^{\frac{1}{2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.7

Move the negative in front of the fraction.

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{(- 9) e^{- \frac{{(- 9)}^{2}}{162}}}{27}$

Step 10.1.8

Move $e^{- \frac{{(- 9)}^{2}}{162}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 9}{27 e^{\frac{{(- 9)}^{2}}{162}}}$

Step 10.1.9

Factor $9$ out of $- 9$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{9 (- 1)}{27 e^{\frac{{(- 9)}^{2}}{162}}}$

Step 10.1.10

Cancel the common factors.

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Step 10.1.10.1

Factor $9$ out of $27 e^{\frac{{(- 9)}^{2}}{162}}$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{9 (- 1)}{9 (3 e^{\frac{{(- 9)}^{2}}{162}})}$

Step 10.1.10.2

Cancel the common factor.

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{9 \cdot - 1}{9 (3 e^{\frac{{(- 9)}^{2}}{162}})}$

Step 10.1.10.3

Rewrite the expression.

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{{(- 9)}^{2}}{162}}}$

Step 10.1.11

Raise $- 9$ to the power of $2$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{81}{162}}}$

Step 10.1.12

Cancel the common factor of $81$ and $162$ .

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Step 10.1.12.1

Factor $81$ out of $81$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{81 (1)}{162}}}$

Step 10.1.12.2

Cancel the common factors.

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Step 10.1.12.2.1

Factor $81$ out of $162$ .

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{81 \cdot 1}{81 \cdot 2}}}$

Step 10.1.12.2.2

Cancel the common factor.

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{81 \cdot 1}{81 \cdot 2}}}$

Step 10.1.12.2.3

Rewrite the expression.

$- \frac{1}{9 e^{\frac{1}{2}}} - \frac{- 1}{3 e^{\frac{1}{2}}}$

Step 10.1.13

Move the negative in front of the fraction.

$- \frac{1}{9 e^{\frac{1}{2}}} - - \frac{1}{3 e^{\frac{1}{2}}}$

Step 10.1.14

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

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Step 10.1.14.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{9 e^{\frac{1}{2}}} + 1 \frac{1}{3 e^{\frac{1}{2}}}$

Step 10.1.14.2

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

$- \frac{1}{9 e^{\frac{1}{2}}} + \frac{1}{3 e^{\frac{1}{2}}}$

Step 10.2

To write $\frac{1}{3 e^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{9 e^{\frac{1}{2}}} + \frac{1}{3 e^{\frac{1}{2}}} \cdot \frac{3}{3}$

Step 10.3

Write each expression with a common denominator of $9 e^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.1

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

$- \frac{1}{9 e^{\frac{1}{2}}} + \frac{3}{3 e^{\frac{1}{2}} \cdot 3}$

Step 10.3.2

Multiply $3$ by $3$ .

$- \frac{1}{9 e^{\frac{1}{2}}} + \frac{3}{9 e^{\frac{1}{2}}}$

Step 10.4

Combine the numerators over the common denominator.

$\frac{- 1 + 3}{9 e^{\frac{1}{2}}}$

Step 10.5

Add $- 1$ and $3$ .

$\frac{2}{9 e^{\frac{1}{2}}}$

Step 11

$x = - 9$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 9$ is a local minimum

Step 12

Find the y-value when $x = - 9$ .

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Step 12.1

Replace the variable $x$ with $- 9$ in the expression.

$f (- 9) = (- 9) \cdot e^{- \frac{{(- 9)}^{2}}{162}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Raise $- 9$ to the power of $2$ .

$f (- 9) = - 9 \cdot e^{- \frac{81}{162}}$

Step 12.2.2

Cancel the common factor of $81$ and $162$ .

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Step 12.2.2.1

Factor $81$ out of $81$ .

$f (- 9) = - 9 \cdot e^{- \frac{81 (1)}{162}}$

Step 12.2.2.2

Cancel the common factors.

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Step 12.2.2.2.1

Factor $81$ out of $162$ .

$f (- 9) = - 9 \cdot e^{- \frac{81 \cdot 1}{81 \cdot 2}}$

Step 12.2.2.2.2

Cancel the common factor.

$f (- 9) = - 9 \cdot e^{- \frac{81 \cdot 1}{81 \cdot 2}}$

Step 12.2.2.2.3

Rewrite the expression.

$f (- 9) = - 9 \cdot e^{- \frac{1}{2}}$

Step 12.2.3

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

$f (- 9) = - 9 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 12.2.4

Combine $- 9$ and $\frac{1}{e^{\frac{1}{2}}}$ .

$f (- 9) = \frac{- 9}{e^{\frac{1}{2}}}$

Step 12.2.5

Move the negative in front of the fraction.

$f (- 9) = - \frac{9}{e^{\frac{1}{2}}}$

Step 12.2.6

The final answer is $- \frac{9}{e^{\frac{1}{2}}}$ .

$y = - \frac{9}{e^{\frac{1}{2}}}$

Step 13

Evaluate the second derivative at $x = 9$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(9)}^{3} e^{- \frac{{(9)}^{2}}{162}}}{6561} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Move $e^{- \frac{{(9)}^{2}}{162}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(9)}^{3}}{6561 e^{\frac{9^{2}}{162}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.2

Raise $9$ to the power of $2$ .

$\frac{9^{3}}{6561 e^{\frac{81}{162}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.3

Raise $9$ to the power of $3$ .

$\frac{729}{6561 e^{\frac{81}{162}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.4

Cancel the common factor of $81$ and $162$ .

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Step 14.1.4.1

Factor $81$ out of $81$ .

$\frac{729}{6561 e^{\frac{81 (1)}{162}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.4.2

Cancel the common factors.

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Step 14.1.4.2.1

Factor $81$ out of $162$ .

$\frac{729}{6561 e^{\frac{81 \cdot 1}{81 \cdot 2}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.4.2.2

Cancel the common factor.

$\frac{729}{6561 e^{\frac{81 \cdot 1}{81 \cdot 2}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.4.2.3

Rewrite the expression.

$\frac{729}{6561 e^{\frac{1}{2}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.5

Factor $729$ out of $729$ .

$\frac{729 (1)}{6561 e^{\frac{1}{2}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.6

Cancel the common factors.

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Step 14.1.6.1

Factor $729$ out of $6561 e^{\frac{1}{2}}$ .

$\frac{729 (1)}{729 (9 e^{\frac{1}{2}})} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.6.2

Cancel the common factor.

$\frac{729 \cdot 1}{729 (9 e^{\frac{1}{2}})} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.6.3

Rewrite the expression.

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{(9) e^{- \frac{{(9)}^{2}}{162}}}{27}$

Step 14.1.7

Move $e^{- \frac{{(9)}^{2}}{162}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{9}{27 e^{\frac{9^{2}}{162}}}$

Step 14.1.8

Factor $9$ out of $9$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{9 (1)}{27 e^{\frac{9^{2}}{162}}}$

Step 14.1.9

Cancel the common factors.

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Step 14.1.9.1

Factor $9$ out of $27 e^{\frac{9^{2}}{162}}$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{9 (1)}{9 (3 e^{\frac{9^{2}}{162}})}$

Step 14.1.9.2

Cancel the common factor.

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{9 \cdot 1}{9 (3 e^{\frac{9^{2}}{162}})}$

Step 14.1.9.3

Rewrite the expression.

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{9^{2}}{162}}}$

Step 14.1.10

Raise $9$ to the power of $2$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{81}{162}}}$

Step 14.1.11

Cancel the common factor of $81$ and $162$ .

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Step 14.1.11.1

Factor $81$ out of $81$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{81 (1)}{162}}}$

Step 14.1.11.2

Cancel the common factors.

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Step 14.1.11.2.1

Factor $81$ out of $162$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{81 \cdot 1}{81 \cdot 2}}}$

Step 14.1.11.2.2

Cancel the common factor.

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{81 \cdot 1}{81 \cdot 2}}}$

Step 14.1.11.2.3

Rewrite the expression.

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{1}{2}}}$

Step 14.2

To write $- \frac{1}{3 e^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{1}{3 e^{\frac{1}{2}}} \cdot \frac{3}{3}$

Step 14.3

Write each expression with a common denominator of $9 e^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.1

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

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{3}{3 e^{\frac{1}{2}} \cdot 3}$

Step 14.3.2

Multiply $3$ by $3$ .

$\frac{1}{9 e^{\frac{1}{2}}} - \frac{3}{9 e^{\frac{1}{2}}}$

Step 14.4

Combine the numerators over the common denominator.

$\frac{1 - 3}{9 e^{\frac{1}{2}}}$

Step 14.5

Subtract $3$ from $1$ .

$\frac{- 2}{9 e^{\frac{1}{2}}}$

Step 14.6

Move the negative in front of the fraction.

$- \frac{2}{9 e^{\frac{1}{2}}}$

Step 15

$x = 9$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 9$ is a local maximum

Step 16

Find the y-value when $x = 9$ .

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Step 16.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = (9) \cdot e^{- \frac{{(9)}^{2}}{162}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Raise $9$ to the power of $2$ .

$f (9) = 9 \cdot e^{- \frac{81}{162}}$

Step 16.2.2

Cancel the common factor of $81$ and $162$ .

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Step 16.2.2.1

Factor $81$ out of $81$ .

$f (9) = 9 \cdot e^{- \frac{81 (1)}{162}}$

Step 16.2.2.2

Cancel the common factors.

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Step 16.2.2.2.1

Factor $81$ out of $162$ .

$f (9) = 9 \cdot e^{- \frac{81 \cdot 1}{81 \cdot 2}}$

Step 16.2.2.2.2

Cancel the common factor.

$f (9) = 9 \cdot e^{- \frac{81 \cdot 1}{81 \cdot 2}}$

Step 16.2.2.2.3

Rewrite the expression.

$f (9) = 9 \cdot e^{- \frac{1}{2}}$

Step 16.2.3

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

$f (9) = 9 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 16.2.4

Combine $9$ and $\frac{1}{e^{\frac{1}{2}}}$ .

$f (9) = \frac{9}{e^{\frac{1}{2}}}$

Step 16.2.5

The final answer is $\frac{9}{e^{\frac{1}{2}}}$ .

$y = \frac{9}{e^{\frac{1}{2}}}$

Step 17

These are the local extrema for $f (x) = x e^{- \frac{x^{2}}{162}}$ .

$(- 9, - \frac{9}{e^{\frac{1}{2}}})$ is a local minima

$(9, \frac{9}{e^{\frac{1}{2}}})$ is a local maxima

Step 18