Calculus Examples

$e^{- \frac{x^{2}}{32}}$

Step 1

Write $e^{- \frac{x^{2}}{32}}$ as a function.

$f (x) = e^{- \frac{x^{2}}{32}}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $g (x) = - \frac{x^{2}}{32}$ .

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Step 2.1.1.1.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{32}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{32}]$

Step 2.1.1.1.2

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

$e^{u} \frac{d}{d x} [- \frac{x^{2}}{32}]$

Step 2.1.1.1.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{32}$ .

$e^{- \frac{x^{2}}{32}} \frac{d}{d x} [- \frac{x^{2}}{32}]$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

Since $- \frac{1}{32}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{32}$ with respect to $x$ is $- \frac{1}{32} \frac{d}{d x} [x^{2}]$ .

$e^{- \frac{x^{2}}{32}} (- \frac{1}{32} \frac{d}{d x} [x^{2}])$

Step 2.1.1.2.2

Combine $e^{- \frac{x^{2}}{32}}$ and $\frac{1}{32}$ .

$- \frac{e^{- \frac{x^{2}}{32}}}{32} \frac{d}{d x} [x^{2}]$

Step 2.1.1.2.3

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

$- \frac{e^{- \frac{x^{2}}{32}}}{32} (2 x)$

Step 2.1.1.2.4

Simplify terms.

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Step 2.1.1.2.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{e^{- \frac{x^{2}}{32}}}{32} x$

Step 2.1.1.2.4.2

Combine $- 2$ and $\frac{e^{- \frac{x^{2}}{32}}}{32}$ .

$\frac{- 2 e^{- \frac{x^{2}}{32}}}{32} x$

Step 2.1.1.2.4.3

Combine $\frac{- 2 e^{- \frac{x^{2}}{32}}}{32}$ and $x$ .

$\frac{- 2 e^{- \frac{x^{2}}{32}} x}{32}$

Step 2.1.1.2.4.4

Cancel the common factor of $- 2$ and $32$ .

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Step 2.1.1.2.4.4.1

Factor $2$ out of $- 2 e^{- \frac{x^{2}}{32}} x$ .

$\frac{2 (- e^{- \frac{x^{2}}{32}} x)}{32}$

Step 2.1.1.2.4.4.2

Cancel the common factors.

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Step 2.1.1.2.4.4.2.1

Factor $2$ out of $32$ .

$\frac{2 (- e^{- \frac{x^{2}}{32}} x)}{2 (16)}$

Step 2.1.1.2.4.4.2.2

Cancel the common factor.

$\frac{2 (- e^{- \frac{x^{2}}{32}} x)}{2 \cdot 16}$

Step 2.1.1.2.4.4.2.3

Rewrite the expression.

$\frac{- e^{- \frac{x^{2}}{32}} x}{16}$

Step 2.1.1.2.4.5

Simplify the expression.

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Step 2.1.1.2.4.5.1

Move the negative in front of the fraction.

$- \frac{e^{- \frac{x^{2}}{32}} x}{16}$

Step 2.1.1.2.4.5.2

Reorder factors in $e^{- \frac{x^{2}}{32}} x$ .

$f' (x) = - \frac{x e^{- \frac{x^{2}}{32}}}{16}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $- \frac{1}{16}$ is constant with respect to $x$ , the derivative of $- \frac{x e^{- \frac{x^{2}}{32}}}{16}$ with respect to $x$ is $- \frac{1}{16} \frac{d}{d x} [x e^{- \frac{x^{2}}{32}}]$ .

$- \frac{1}{16} \frac{d}{d x} [x e^{- \frac{x^{2}}{32}}]$

Step 2.1.2.2

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 $f (x) = x$ and $g (x) = e^{- \frac{x^{2}}{32}}$ .

$- \frac{1}{16} (x \frac{d}{d x} [e^{- \frac{x^{2}}{32}}] + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.3

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 $g (x) = - \frac{x^{2}}{32}$ .

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Step 2.1.2.3.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{32}$ .

$- \frac{1}{16} (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{32}]) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.3.2

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

$- \frac{1}{16} (x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{32}]) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.3.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{32}$ .

$- \frac{1}{16} (x (e^{- \frac{x^{2}}{32}} \frac{d}{d x} [- \frac{x^{2}}{32}]) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4

Differentiate.

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Step 2.1.2.4.1

Since $- \frac{1}{32}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{32}$ with respect to $x$ is $- \frac{1}{32} \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{16} (x (e^{- \frac{x^{2}}{32}} (- \frac{1}{32} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.2

Combine fractions.

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Step 2.1.2.4.2.1

Combine $e^{- \frac{x^{2}}{32}}$ and $\frac{1}{32}$ .

$- \frac{1}{16} (x (- \frac{e^{- \frac{x^{2}}{32}}}{32} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{32}}}{32}$ .

$- \frac{1}{16} (- \frac{x e^{- \frac{x^{2}}{32}}}{32} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.3

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

$- \frac{1}{16} (- \frac{x e^{- \frac{x^{2}}{32}}}{32} (2 x) + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.4

Combine fractions.

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Step 2.1.2.4.4.1

Multiply $2$ by $- 1$ .

$- \frac{1}{16} (- 2 \frac{x e^{- \frac{x^{2}}{32}}}{32} x + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{32}}}{32}$ .

$- \frac{1}{16} (\frac{- 2 (x e^{- \frac{x^{2}}{32}})}{32} x + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.4.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{32}})}{32}$ and $x$ .

$- \frac{1}{16} (\frac{- 2 (x e^{- \frac{x^{2}}{32}}) x}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.5

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

$- \frac{1}{16} (\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{32}})}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.6

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

$- \frac{1}{16} (\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{32}})}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.7

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

$- \frac{1}{16} (\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{32}})}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8

Reduce the expression by cancelling the common factors.

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Step 2.1.2.8.1

Add $1$ and $1$ .

$- \frac{1}{16} (\frac{- 2 (x^{2} e^{- \frac{x^{2}}{32}})}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8.2

Cancel the common factor of $- 2$ and $32$ .

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Step 2.1.2.8.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{32}}$ .

$- \frac{1}{16} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{32}})}{32} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8.2.2

Cancel the common factors.

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Step 2.1.2.8.2.2.1

Factor $2$ out of $32$ .

$- \frac{1}{16} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{32}})}{2 (16)} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8.2.2.2

Cancel the common factor.

$- \frac{1}{16} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{32}})}{2 \cdot 16} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8.2.2.3

Rewrite the expression.

$- \frac{1}{16} (\frac{- x^{2} e^{- \frac{x^{2}}{32}}}{16} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.8.3

Move the negative in front of the fraction.

$- \frac{1}{16} (- \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16} + e^{- \frac{x^{2}}{32}} \frac{d}{d x} [x])$

Step 2.1.2.9

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

$- \frac{1}{16} (- \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16} + e^{- \frac{x^{2}}{32}} \cdot 1)$

Step 2.1.2.10

Multiply $e^{- \frac{x^{2}}{32}}$ by $1$ .

$- \frac{1}{16} (- \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16} + e^{- \frac{x^{2}}{32}})$

Step 2.1.2.11

Simplify.

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Step 2.1.2.11.1

Apply the distributive property.

$- \frac{1}{16} (- \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16}) - \frac{1}{16} e^{- \frac{x^{2}}{32}}$

Step 2.1.2.11.2

Combine terms.

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Step 2.1.2.11.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{16}) \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16} - \frac{1}{16} e^{- \frac{x^{2}}{32}}$

Step 2.1.2.11.2.2

Multiply $\frac{1}{16}$ by $1$ .

$\frac{1}{16} \cdot \frac{x^{2} e^{- \frac{x^{2}}{32}}}{16} - \frac{1}{16} e^{- \frac{x^{2}}{32}}$

Step 2.1.2.11.2.3

Multiply $\frac{1}{16}$ by $\frac{x^{2} e^{- \frac{x^{2}}{32}}}{16}$ .

$\frac{x^{2} e^{- \frac{x^{2}}{32}}}{16 \cdot 16} - \frac{1}{16} e^{- \frac{x^{2}}{32}}$

Step 2.1.2.11.2.4

Multiply $16$ by $16$ .

$\frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{1}{16} e^{- \frac{x^{2}}{32}}$

Step 2.1.2.11.2.5

Combine $e^{- \frac{x^{2}}{32}}$ and $\frac{1}{16}$ .

$f'' (x) = \frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{e^{- \frac{x^{2}}{32}}}{16}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{e^{- \frac{x^{2}}{32}}}{16}$ .

$\frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{e^{- \frac{x^{2}}{32}}}{16}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{e^{- \frac{x^{2}}{32}}}{16} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{x^{2} e^{- \frac{x^{2}}{32}}}{256} - \frac{e^{- \frac{x^{2}}{32}}}{16} = 0$

Step 2.2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 4, 4$

Step 3

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

$(- \infty, - 4) \cup (- 4, 4) \cup (4, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 6$ in the expression.

$f'' (- 6) = \frac{{(- 6)}^{2} e^{- \frac{{(- 6)}^{2}}{32}}}{256} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Move $e^{- \frac{{(- 6)}^{2}}{32}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 6) = \frac{{(- 6)}^{2}}{256 e^{\frac{{(- 6)}^{2}}{32}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.2

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = \frac{{(- 6)}^{2}}{256 e^{\frac{36}{32}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.3

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = \frac{36}{256 e^{\frac{36}{32}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.4

Cancel the common factor of $36$ and $32$ .

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Step 5.2.1.4.1

Factor $4$ out of $36$ .

$f'' (- 6) = \frac{36}{256 e^{\frac{4 (9)}{32}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.4.2

Cancel the common factors.

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Step 5.2.1.4.2.1

Factor $4$ out of $32$ .

$f'' (- 6) = \frac{36}{256 e^{\frac{4 \cdot 9}{4 \cdot 8}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.4.2.2

Cancel the common factor.

$f'' (- 6) = \frac{36}{256 e^{\frac{4 \cdot 9}{4 \cdot 8}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.4.2.3

Rewrite the expression.

$f'' (- 6) = \frac{36}{256 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.5

Factor $4$ out of $36$ .

$f'' (- 6) = \frac{4 (9)}{256 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.6

Cancel the common factors.

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Step 5.2.1.6.1

Factor $4$ out of $256 e^{\frac{9}{8}}$ .

$f'' (- 6) = \frac{4 (9)}{4 (64 e^{\frac{9}{8}})} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.6.2

Cancel the common factor.

$f'' (- 6) = \frac{4 \cdot 9}{4 (64 e^{\frac{9}{8}})} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.6.3

Rewrite the expression.

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(- 6)}^{2}}{32}}}{16}$

Step 5.2.1.7

Move $e^{- \frac{{(- 6)}^{2}}{32}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{{(- 6)}^{2}}{32}}}$

Step 5.2.1.8

Raise $- 6$ to the power of $2$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{36}{32}}}$

Step 5.2.1.9

Cancel the common factor of $36$ and $32$ .

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Step 5.2.1.9.1

Factor $4$ out of $36$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 (9)}{32}}}$

Step 5.2.1.9.2

Cancel the common factors.

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Step 5.2.1.9.2.1

Factor $4$ out of $32$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 \cdot 9}{4 \cdot 8}}}$

Step 5.2.1.9.2.2

Cancel the common factor.

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 \cdot 9}{4 \cdot 8}}}$

Step 5.2.1.9.2.3

Rewrite the expression.

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{9}{8}}}$

Step 5.2.2

To write $- \frac{1}{16 e^{\frac{9}{8}}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{9}{8}}} \cdot \frac{4}{4}$

Step 5.2.3

Write each expression with a common denominator of $64 e^{\frac{9}{8}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1

Multiply $\frac{1}{16 e^{\frac{9}{8}}}$ by $\frac{4}{4}$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{4}{16 e^{\frac{9}{8}} \cdot 4}$

Step 5.2.3.2

Multiply $4$ by $16$ .

$f'' (- 6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{4}{64 e^{\frac{9}{8}}}$

Step 5.2.4

Combine the numerators over the common denominator.

$f'' (- 6) = \frac{9 - 4}{64 e^{\frac{9}{8}}}$

Step 5.2.5

Subtract $4$ from $9$ .

$f'' (- 6) = \frac{5}{64 e^{\frac{9}{8}}}$

Step 5.2.6

The final answer is $\frac{5}{64 e^{\frac{9}{8}}}$ .

$\frac{5}{64 e^{\frac{9}{8}}}$

Step 5.3

The graph is concave up on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 4, 4)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (0) = \frac{{(0)}^{2} e^{- \frac{{(0)}^{2}}{32}}}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f'' (0) = \frac{0^{2} e^{- \frac{0}{32}}}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.2

Simplify the numerator.

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Step 6.2.1.2.1

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

$f'' (0) = \frac{0 e^{- \frac{0}{32}}}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.2.2

Divide $0$ by $32$ .

$f'' (0) = \frac{0 e^{- 0}}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.2.3

Multiply $- 1$ by $0$ .

$f'' (0) = \frac{0 e^{0}}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.2.4

Anything raised to $0$ is $1$ .

$f'' (0) = \frac{0 \cdot 1}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.3

Multiply $0$ by $1$ .

$f'' (0) = \frac{0}{256} - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.4

Divide $0$ by $256$ .

$f'' (0) = 0 - \frac{e^{- \frac{{(0)}^{2}}{32}}}{16}$

Step 6.2.1.5

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

$f'' (0) = 0 - \frac{e^{- \frac{0}{32}}}{16}$

Step 6.2.1.6

Simplify the numerator.

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Step 6.2.1.6.1

Divide $0$ by $32$ .

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

Step 6.2.1.6.2

Multiply $- 1$ by $0$ .

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

Step 6.2.1.6.3

Anything raised to $0$ is $1$ .

$f'' (0) = 0 - \frac{1}{16}$

Step 6.2.2

Subtract $\frac{1}{16}$ from $0$ .

$f'' (0) = - \frac{1}{16}$

Step 6.2.3

The final answer is $- \frac{1}{16}$ .

$- \frac{1}{16}$

Step 6.3

The graph is concave down on the interval $(- 4, 4)$ because $f'' (0)$ is negative.

Concave down on $(- 4, 4)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(4, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $6$ in the expression.

$f'' (6) = \frac{{(6)}^{2} e^{- \frac{{(6)}^{2}}{32}}}{256} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Move $e^{- \frac{{(6)}^{2}}{32}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (6) = \frac{{(6)}^{2}}{256 e^{\frac{6^{2}}{32}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.2

Raise $6$ to the power of $2$ .

$f'' (6) = \frac{6^{2}}{256 e^{\frac{36}{32}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.3

Raise $6$ to the power of $2$ .

$f'' (6) = \frac{36}{256 e^{\frac{36}{32}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.4

Cancel the common factor of $36$ and $32$ .

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Step 7.2.1.4.1

Factor $4$ out of $36$ .

$f'' (6) = \frac{36}{256 e^{\frac{4 (9)}{32}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.4.2

Cancel the common factors.

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Step 7.2.1.4.2.1

Factor $4$ out of $32$ .

$f'' (6) = \frac{36}{256 e^{\frac{4 \cdot 9}{4 \cdot 8}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.4.2.2

Cancel the common factor.

$f'' (6) = \frac{36}{256 e^{\frac{4 \cdot 9}{4 \cdot 8}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.4.2.3

Rewrite the expression.

$f'' (6) = \frac{36}{256 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.5

Factor $4$ out of $36$ .

$f'' (6) = \frac{4 (9)}{256 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.6

Cancel the common factors.

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Step 7.2.1.6.1

Factor $4$ out of $256 e^{\frac{9}{8}}$ .

$f'' (6) = \frac{4 (9)}{4 (64 e^{\frac{9}{8}})} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.6.2

Cancel the common factor.

$f'' (6) = \frac{4 \cdot 9}{4 (64 e^{\frac{9}{8}})} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.6.3

Rewrite the expression.

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{e^{- \frac{{(6)}^{2}}{32}}}{16}$

Step 7.2.1.7

Move $e^{- \frac{{(6)}^{2}}{32}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{6^{2}}{32}}}$

Step 7.2.1.8

Raise $6$ to the power of $2$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{36}{32}}}$

Step 7.2.1.9

Cancel the common factor of $36$ and $32$ .

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Step 7.2.1.9.1

Factor $4$ out of $36$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 (9)}{32}}}$

Step 7.2.1.9.2

Cancel the common factors.

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Step 7.2.1.9.2.1

Factor $4$ out of $32$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 \cdot 9}{4 \cdot 8}}}$

Step 7.2.1.9.2.2

Cancel the common factor.

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{4 \cdot 9}{4 \cdot 8}}}$

Step 7.2.1.9.2.3

Rewrite the expression.

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{9}{8}}}$

Step 7.2.2

To write $- \frac{1}{16 e^{\frac{9}{8}}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{1}{16 e^{\frac{9}{8}}} \cdot \frac{4}{4}$

Step 7.2.3

Write each expression with a common denominator of $64 e^{\frac{9}{8}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.1

Multiply $\frac{1}{16 e^{\frac{9}{8}}}$ by $\frac{4}{4}$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{4}{16 e^{\frac{9}{8}} \cdot 4}$

Step 7.2.3.2

Multiply $4$ by $16$ .

$f'' (6) = \frac{9}{64 e^{\frac{9}{8}}} - \frac{4}{64 e^{\frac{9}{8}}}$

Step 7.2.4

Combine the numerators over the common denominator.

$f'' (6) = \frac{9 - 4}{64 e^{\frac{9}{8}}}$

Step 7.2.5

Subtract $4$ from $9$ .

$f'' (6) = \frac{5}{64 e^{\frac{9}{8}}}$

Step 7.2.6

The final answer is $\frac{5}{64 e^{\frac{9}{8}}}$ .

$\frac{5}{64 e^{\frac{9}{8}}}$

Step 7.3

The graph is concave up on the interval $(4, \infty)$ because $f'' (6)$ is positive.

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 8

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

Concave up on $(- \infty, - 4)$ since $f'' (x)$ is positive

Concave down on $(- 4, 4)$ since $f'' (x)$ is negative

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 9