Calculus Examples

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

Step 1

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

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

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

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

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

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

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

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

Step 2.1.1.3

Differentiate.

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Combine fractions.

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Step 2.1.1.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.3.2.2

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

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

Step 2.1.1.3.2.3

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

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

Step 2.1.1.3.3

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

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

Step 2.1.1.3.4

Simplify terms.

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Step 2.1.1.3.4.1

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

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

Step 2.1.1.3.4.2

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

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

Step 2.1.1.3.4.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.3.1

Cancel the common factor.

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

Step 2.1.1.3.4.3.2

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

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

Step 2.1.1.3.4.4

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

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

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

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

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

Combine fractions.

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Step 2.1.2.3.2.1

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

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

Step 2.1.2.3.2.2

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

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

Step 2.1.2.3.3

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

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

Step 2.1.2.3.4

Combine fractions.

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Step 2.1.2.3.4.1

Multiply $2$ by $- 1$ .

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

Step 2.1.2.3.4.2

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

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

Step 2.1.2.3.4.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

Reduce the expression by cancelling the common factors.

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Step 2.1.2.7.1

Add $1$ and $1$ .

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

Step 2.1.2.7.2

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

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Step 2.1.2.7.2.1

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

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

Step 2.1.2.7.2.2

Cancel the common factors.

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Step 2.1.2.7.2.2.1

Factor $2$ out of $2$ .

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

Step 2.1.2.7.2.2.2

Cancel the common factor.

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

Step 2.1.2.7.2.2.3

Rewrite the expression.

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

Step 2.1.2.7.2.2.4

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

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

Step 2.1.2.8

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

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

Step 2.1.2.9

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

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

Multiply by $1$ .

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

Step 2.2.2.1.3

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 2.2.2.4

Factor.

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Step 2.2.2.4.1

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 $b = x$ .

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

Step 2.2.2.4.2

Remove unnecessary parentheses.

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

Step 2.2.3

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

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

$1 + x = 0$

$1 - x = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

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

Step 2.2.4.2.2

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

Undefined

Step 2.2.4.2.3

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

No solution

Step 2.2.5

Set $1 + x$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $1 + x$ equal to $0$ .

$1 + x = 0$

Step 2.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.6

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 2.2.6.2

Solve $1 - x = 0$ for $x$ .

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Step 2.2.6.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 2.2.6.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 2.2.6.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

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

Step 2.2.6.2.2.2

Simplify the left side.

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Step 2.2.6.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.6.2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.6.2.2.3

Simplify the right side.

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Step 2.2.6.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 2.2.7

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

$x = - 1, 1$

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.

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

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

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

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

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

Step 5.2.1.2

Multiply $- 1$ by $16$ .

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Divide $16$ by $2$ .

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

Step 5.2.1.5

Multiply $- 1$ by $8$ .

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

Step 5.2.1.6

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

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

Step 5.2.1.7

Combine $- 16$ and $\frac{1}{e^{8}}$ .

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

Step 5.2.1.8

Move the negative in front of the fraction.

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

Step 5.2.1.9

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

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

Step 5.2.1.10

Divide $16$ by $2$ .

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

Step 5.2.1.11

Multiply $- 1$ by $8$ .

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

Step 5.2.1.12

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

$f'' (- 4) = - \frac{16}{e^{8}} + \frac{1}{e^{8}}$

Step 5.2.2

Combine fractions.

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Step 5.2.2.1

Combine the numerators over the common denominator.

$f'' (- 4) = \frac{- 16 + 1}{e^{8}}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Add $- 16$ and $1$ .

$f'' (- 4) = \frac{- 15}{e^{8}}$

Step 5.2.2.2.2

Move the negative in front of the fraction.

$f'' (- 4) = - \frac{15}{e^{8}}$

Step 5.2.3

The final answer is $- \frac{15}{e^{8}}$ .

$- \frac{15}{e^{8}}$

Step 5.3

The graph is concave down on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is negative.

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- 1, 1)$ 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.

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

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 $0$ .

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

Step 6.2.1.2

Multiply $- 1$ by $0$ .

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Divide $0$ by $2$ .

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

Step 6.2.1.5

Multiply $- 1$ by $0$ .

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

Step 6.2.1.6

Anything raised to $0$ is $1$ .

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

Step 6.2.1.7

Multiply $0$ by $1$ .

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

Step 6.2.1.8

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

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

Step 6.2.1.9

Divide $0$ by $2$ .

$f'' (0) = 0 + e^{- 0}$

Step 6.2.1.10

Multiply $- 1$ by $0$ .

$f'' (0) = 0 + e^{0}$

Step 6.2.1.11

Anything raised to $0$ is $1$ .

$f'' (0) = 0 + 1$

Step 6.2.2

Add $0$ and $1$ .

$f'' (0) = 1$

Step 6.2.3

The final answer is $1$ .

$1$

Step 6.3

The graph is concave up on the interval $(- 1, 1)$ because $f'' (0)$ is positive.

Concave up on $(- 1, 1)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $4$ in the expression.

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

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

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

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

Step 7.2.1.2

Multiply $- 1$ by $16$ .

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Divide $16$ by $2$ .

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

Step 7.2.1.5

Multiply $- 1$ by $8$ .

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

Step 7.2.1.6

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

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

Step 7.2.1.7

Combine $- 16$ and $\frac{1}{e^{8}}$ .

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

Step 7.2.1.8

Move the negative in front of the fraction.

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

Step 7.2.1.9

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

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

Step 7.2.1.10

Divide $16$ by $2$ .

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

Step 7.2.1.11

Multiply $- 1$ by $8$ .

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

Step 7.2.1.12

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

$f'' (4) = - \frac{16}{e^{8}} + \frac{1}{e^{8}}$

Step 7.2.2

Combine fractions.

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Step 7.2.2.1

Combine the numerators over the common denominator.

$f'' (4) = \frac{- 16 + 1}{e^{8}}$

Step 7.2.2.2

Simplify the expression.

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Step 7.2.2.2.1

Add $- 16$ and $1$ .

$f'' (4) = \frac{- 15}{e^{8}}$

Step 7.2.2.2.2

Move the negative in front of the fraction.

$f'' (4) = - \frac{15}{e^{8}}$

Step 7.2.3

The final answer is $- \frac{15}{e^{8}}$ .

$- \frac{15}{e^{8}}$

Step 7.3

The graph is concave down on the interval $(1, \infty)$ because $f'' (4)$ is negative.

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 8

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

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Concave up on $(- 1, 1)$ since $f'' (x)$ is positive

Concave down on $(1, \infty)$ since $f'' (x)$ is negative

Step 9