Calculus Examples

$f (x) = - e^{- \frac{x^{2}}{50}}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{- \frac{x^{2}}{50}}$ with respect to $x$ is $- \frac{d}{d x} [e^{- \frac{x^{2}}{50}}]$ .

$- \frac{d}{d x} [e^{- \frac{x^{2}}{50}}]$

Step 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 $g (x) = - \frac{x^{2}}{50}$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{50}$ .

$- (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{50}])$

Step 1.1.2.2

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

$- (e^{u} \frac{d}{d x} [- \frac{x^{2}}{50}])$

Step 1.1.2.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{50}$ .

$- (e^{- \frac{x^{2}}{50}} \frac{d}{d x} [- \frac{x^{2}}{50}])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $- \frac{1}{50}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{50}$ with respect to $x$ is $- \frac{1}{50} \frac{d}{d x} [x^{2}]$ .

$- e^{- \frac{x^{2}}{50}} (- \frac{1}{50} \frac{d}{d x} [x^{2}])$

Step 1.1.3.2

Combine fractions.

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Step 1.1.3.2.1

Multiply $- 1$ by $- 1$ .

$1 e^{- \frac{x^{2}}{50}} (\frac{1}{50} \frac{d}{d x} [x^{2}])$

Step 1.1.3.2.2

Multiply $e^{- \frac{x^{2}}{50}}$ by $1$ .

$e^{- \frac{x^{2}}{50}} (\frac{1}{50} \frac{d}{d x} [x^{2}])$

Step 1.1.3.2.3

Combine $\frac{1}{50}$ and $e^{- \frac{x^{2}}{50}}$ .

$\frac{e^{- \frac{x^{2}}{50}}}{50} \frac{d}{d x} [x^{2}]$

Step 1.1.3.3

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

$\frac{e^{- \frac{x^{2}}{50}}}{50} (2 x)$

Step 1.1.3.4

Simplify terms.

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Step 1.1.3.4.1

Combine $2$ and $\frac{e^{- \frac{x^{2}}{50}}}{50}$ .

$\frac{2 e^{- \frac{x^{2}}{50}}}{50} x$

Step 1.1.3.4.2

Combine $\frac{2 e^{- \frac{x^{2}}{50}}}{50}$ and $x$ .

$\frac{2 e^{- \frac{x^{2}}{50}} x}{50}$

Step 1.1.3.4.3

Cancel the common factor of $2$ and $50$ .

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Step 1.1.3.4.3.1

Factor $2$ out of $2 e^{- \frac{x^{2}}{50}} x$ .

$\frac{2 (e^{- \frac{x^{2}}{50}} x)}{50}$

Step 1.1.3.4.3.2

Cancel the common factors.

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Step 1.1.3.4.3.2.1

Factor $2$ out of $50$ .

$\frac{2 (e^{- \frac{x^{2}}{50}} x)}{2 \cdot 25}$

Step 1.1.3.4.3.2.2

Cancel the common factor.

$\frac{2 (e^{- \frac{x^{2}}{50}} x)}{2 \cdot 25}$

Step 1.1.3.4.3.2.3

Rewrite the expression.

$\frac{e^{- \frac{x^{2}}{50}} x}{25}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Reorder factors in $e^{- \frac{x^{2}}{50}} x$ .

$\frac{x e^{- \frac{x^{2}}{50}}}{25}$

Step 1.1.4.2

Reorder terms.

$\frac{e^{- \frac{x^{2}}{50}} x}{25}$

Step 1.1.4.3

Reorder factors in $\frac{e^{- \frac{x^{2}}{50}} x}{25}$ .

$f' (x) = \frac{x e^{- \frac{x^{2}}{50}}}{25}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $\frac{1}{25}$ is constant with respect to $x$ , the derivative of $\frac{x e^{- \frac{x^{2}}{50}}}{25}$ with respect to $x$ is $\frac{1}{25} \frac{d}{d x} [x e^{- \frac{x^{2}}{50}}]$ .

$\frac{1}{25} \frac{d}{d x} [x e^{- \frac{x^{2}}{50}}]$

Step 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 $g (x) = e^{- \frac{x^{2}}{50}}$ .

$\frac{1}{25} (x \frac{d}{d x} [e^{- \frac{x^{2}}{50}}] + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 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 $g (x) = - \frac{x^{2}}{50}$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{50}$ .

$\frac{1}{25} (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{50}]) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.3.2

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

$\frac{1}{25} (x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{50}]) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.3.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{50}$ .

$\frac{1}{25} (x (e^{- \frac{x^{2}}{50}} \frac{d}{d x} [- \frac{x^{2}}{50}]) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4

Differentiate.

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Step 1.2.4.1

Since $- \frac{1}{50}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{50}$ with respect to $x$ is $- \frac{1}{50} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{25} (x (e^{- \frac{x^{2}}{50}} (- \frac{1}{50} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.2

Combine fractions.

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Step 1.2.4.2.1

Combine $e^{- \frac{x^{2}}{50}}$ and $\frac{1}{50}$ .

$\frac{1}{25} (x (- \frac{e^{- \frac{x^{2}}{50}}}{50} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{50}}}{50}$ .

$\frac{1}{25} (- \frac{x e^{- \frac{x^{2}}{50}}}{50} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.3

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

$\frac{1}{25} (- \frac{x e^{- \frac{x^{2}}{50}}}{50} (2 x) + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.4

Combine fractions.

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Step 1.2.4.4.1

Multiply $2$ by $- 1$ .

$\frac{1}{25} (- 2 \frac{x e^{- \frac{x^{2}}{50}}}{50} x + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{50}}}{50}$ .

$\frac{1}{25} (\frac{- 2 (x e^{- \frac{x^{2}}{50}})}{50} x + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.4.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{50}})}{50}$ and $x$ .

$\frac{1}{25} (\frac{- 2 (x e^{- \frac{x^{2}}{50}}) x}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.5

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

$\frac{1}{25} (\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{50}})}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.6

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

$\frac{1}{25} (\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{50}})}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.7

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

$\frac{1}{25} (\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{50}})}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8

Reduce the expression by cancelling the common factors.

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Step 1.2.8.1

Add $1$ and $1$ .

$\frac{1}{25} (\frac{- 2 (x^{2} e^{- \frac{x^{2}}{50}})}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8.2

Cancel the common factor of $- 2$ and $50$ .

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Step 1.2.8.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{50}}$ .

$\frac{1}{25} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{50}})}{50} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8.2.2

Cancel the common factors.

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Step 1.2.8.2.2.1

Factor $2$ out of $50$ .

$\frac{1}{25} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{50}})}{2 (25)} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8.2.2.2

Cancel the common factor.

$\frac{1}{25} (\frac{2 (- x^{2} e^{- \frac{x^{2}}{50}})}{2 \cdot 25} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8.2.2.3

Rewrite the expression.

$\frac{1}{25} (\frac{- x^{2} e^{- \frac{x^{2}}{50}}}{25} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.8.3

Move the negative in front of the fraction.

$\frac{1}{25} (- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{25} + e^{- \frac{x^{2}}{50}} \frac{d}{d x} [x])$

Step 1.2.9

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

$\frac{1}{25} (- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{25} + e^{- \frac{x^{2}}{50}} \cdot 1)$

Step 1.2.10

Multiply $e^{- \frac{x^{2}}{50}}$ by $1$ .

$\frac{1}{25} (- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{25} + e^{- \frac{x^{2}}{50}})$

Step 1.2.11

Simplify.

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Step 1.2.11.1

Apply the distributive property.

$\frac{1}{25} (- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{25}) + \frac{1}{25} e^{- \frac{x^{2}}{50}}$

Step 1.2.11.2

Combine terms.

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Step 1.2.11.2.1

Multiply $\frac{1}{25}$ by $\frac{x^{2} e^{- \frac{x^{2}}{50}}}{25}$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{25 \cdot 25} + \frac{1}{25} e^{- \frac{x^{2}}{50}}$

Step 1.2.11.2.2

Multiply $25$ by $25$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{1}{25} e^{- \frac{x^{2}}{50}}$

Step 1.2.11.2.3

Combine $\frac{1}{25}$ and $e^{- \frac{x^{2}}{50}}$ .

$f'' (x) = - \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{e^{- \frac{x^{2}}{50}}}{25}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{e^{- \frac{x^{2}}{50}}}{25}$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{e^{- \frac{x^{2}}{50}}}{25}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{e^{- \frac{x^{2}}{50}}}{25} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{50}}}{625} + \frac{e^{- \frac{x^{2}}{50}}}{25} = 0$

Step 2.2

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

$x = - 5, 5$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- 5$ in $f (x) = - e^{- \frac{x^{2}}{50}}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - e^{- \frac{{(- 5)}^{2}}{50}}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Raise $- 5$ to the power of $2$ .

$f (- 5) = - e^{- \frac{25}{50}}$

Step 3.1.2.2

Cancel the common factor of $25$ and $50$ .

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Step 3.1.2.2.1

Factor $25$ out of $25$ .

$f (- 5) = - e^{- \frac{25 (1)}{50}}$

Step 3.1.2.2.2

Cancel the common factors.

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Step 3.1.2.2.2.1

Factor $25$ out of $50$ .

$f (- 5) = - e^{- \frac{25 \cdot 1}{25 \cdot 2}}$

Step 3.1.2.2.2.2

Cancel the common factor.

$f (- 5) = - e^{- \frac{25 \cdot 1}{25 \cdot 2}}$

Step 3.1.2.2.2.3

Rewrite the expression.

$f (- 5) = - e^{- \frac{1}{2}}$

Step 3.1.2.3

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

$f (- 5) = - \frac{1}{e^{\frac{1}{2}}}$

Step 3.1.2.4

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

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

Step 3.2

The point found by substituting $- 5$ in $f (x) = - e^{- \frac{x^{2}}{50}}$ is $(- 5, - \frac{1}{e^{\frac{1}{2}}})$ . This point can be an inflection point.

$(- 5, - \frac{1}{e^{\frac{1}{2}}})$

Step 3.3

Substitute $5$ in $f (x) = - e^{- \frac{x^{2}}{50}}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - e^{- \frac{{(5)}^{2}}{50}}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Raise $5$ to the power of $2$ .

$f (5) = - e^{- \frac{25}{50}}$

Step 3.3.2.2

Cancel the common factor of $25$ and $50$ .

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Step 3.3.2.2.1

Factor $25$ out of $25$ .

$f (5) = - e^{- \frac{25 (1)}{50}}$

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor $25$ out of $50$ .

$f (5) = - e^{- \frac{25 \cdot 1}{25 \cdot 2}}$

Step 3.3.2.2.2.2

Cancel the common factor.

$f (5) = - e^{- \frac{25 \cdot 1}{25 \cdot 2}}$

Step 3.3.2.2.2.3

Rewrite the expression.

$f (5) = - e^{- \frac{1}{2}}$

Step 3.3.2.3

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

$f (5) = - \frac{1}{e^{\frac{1}{2}}}$

Step 3.3.2.4

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

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

Step 3.4

The point found by substituting $5$ in $f (x) = - e^{- \frac{x^{2}}{50}}$ is $(5, - \frac{1}{e^{\frac{1}{2}}})$ . This point can be an inflection point.

$(5, - \frac{1}{e^{\frac{1}{2}}})$

Step 3.5

Determine the points that could be inflection points.

$(- 5, - \frac{1}{e^{\frac{1}{2}}}), (5, - \frac{1}{e^{\frac{1}{2}}})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 5) \cup (- 5, 5) \cup (5, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 5)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 5.1$ in the expression.

$f'' (- 5.1) = - \frac{{(- 5.1)}^{2} e^{- \frac{{(- 5.1)}^{2}}{50}}}{625} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

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 $e^{- \frac{{(- 5.1)}^{2}}{50}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 5.1) = - \frac{{(- 5.1)}^{2}}{625 e^{\frac{{(- 5.1)}^{2}}{50}}} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.2

Raise $- 5.1$ to the power of $2$ .

$f'' (- 5.1) = - \frac{{(- 5.1)}^{2}}{625 e^{\frac{26.01}{50}}} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.3

Raise $- 5.1$ to the power of $2$ .

$f'' (- 5.1) = - \frac{26.01}{625 e^{\frac{26.01}{50}}} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.4

Divide $26.01$ by $50$ .

$f'' (- 5.1) = - \frac{26.01}{625 e^{0.5202}} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.5

Replace $e$ with an approximation.

$f'' (- 5.1) = - \frac{26.01}{625 \cdot {2.71828182}^{0.5202}} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.6

Raise $2.71828182$ to the power of $0.5202$ .

$f'' (- 5.1) = - \frac{26.01}{625 \cdot 1.68236408} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.7

Multiply $625$ by $1.68236408$ .

$f'' (- 5.1) = - \frac{26.01}{1051.47755554} + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.8

Divide $26.01$ by $1051.47755554$ .

$f'' (- 5.1) = - 1 \cdot 0.02473661 + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.9

Multiply $- 1$ by $0.02473661$ .

$f'' (- 5.1) = - 0.02473661 + \frac{e^{- \frac{{(- 5.1)}^{2}}{50}}}{25}$

Step 5.2.1.10

Move $e^{- \frac{{(- 5.1)}^{2}}{50}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 5.1) = - 0.02473661 + \frac{1}{25 e^{\frac{{(- 5.1)}^{2}}{50}}}$

Step 5.2.1.11

Raise $- 5.1$ to the power of $2$ .

$f'' (- 5.1) = - 0.02473661 + \frac{1}{25 e^{\frac{26.01}{50}}}$

Step 5.2.1.12

Divide $26.01$ by $50$ .

$f'' (- 5.1) = - 0.02473661 + \frac{1}{25 e^{0.5202}}$

Step 5.2.2

Add $- 0.02473661$ and $\frac{1}{25 e^{0.5202}}$ .

$f'' (- 5.1) = - 0.00096055$

Step 5.2.3

The final answer is $- 0.00096055$ .

$- 0.00096055$

Step 5.3

At $- 5.1$ , the second derivative is $- 0.00096055$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 5)$

Decreasing on $(- \infty, - 5)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 5, 5)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (0) = - \frac{{(0)}^{2} e^{- \frac{{(0)}^{2}}{50}}}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

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 $0$ .

$f'' (0) = - \frac{0^{2} e^{- \frac{0}{50}}}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

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 $0$ .

$f'' (0) = - \frac{0 e^{- \frac{0}{50}}}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.2.2

Divide $0$ by $50$ .

$f'' (0) = - \frac{0 e^{- 0}}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.2.3

Multiply $- 1$ by $0$ .

$f'' (0) = - \frac{0 e^{0}}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.2.4

Anything raised to $0$ is $1$ .

$f'' (0) = - \frac{0 \cdot 1}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.3

Multiply $0$ by $1$ .

$f'' (0) = - \frac{0}{625} + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.4

Divide $0$ by $625$ .

$f'' (0) = - 0 + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.5

Multiply $- 1$ by $0$ .

$f'' (0) = 0 + \frac{e^{- \frac{{(0)}^{2}}{50}}}{25}$

Step 6.2.1.6

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

$f'' (0) = 0 + \frac{e^{- \frac{0}{50}}}{25}$

Step 6.2.1.7

Simplify the numerator.

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Step 6.2.1.7.1

Divide $0$ by $50$ .

$f'' (0) = 0 + \frac{e^{- 0}}{25}$

Step 6.2.1.7.2

Multiply $- 1$ by $0$ .

$f'' (0) = 0 + \frac{e^{0}}{25}$

Step 6.2.1.7.3

Anything raised to $0$ is $1$ .

$f'' (0) = 0 + \frac{1}{25}$

Step 6.2.2

Add $0$ and $\frac{1}{25}$ .

$f'' (0) = \frac{1}{25}$

Step 6.2.3

The final answer is $\frac{1}{25}$ .

$\frac{1}{25}$

Step 6.3

At $0$ , the second derivative is $0.04$ . Since this is positive, the second derivative is increasing on the interval $(- 5, 5)$ .

Increasing on $(- 5, 5)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(5, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $5.1$ in the expression.

$f'' (5.1) = - \frac{{(5.1)}^{2} e^{- \frac{{(5.1)}^{2}}{50}}}{625} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

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 $e^{- \frac{{(5.1)}^{2}}{50}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (5.1) = - \frac{{(5.1)}^{2}}{625 e^{\frac{{5.1}^{2}}{50}}} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.2

Raise $5.1$ to the power of $2$ .

$f'' (5.1) = - \frac{{5.1}^{2}}{625 e^{\frac{26.01}{50}}} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.3

Raise $5.1$ to the power of $2$ .

$f'' (5.1) = - \frac{26.01}{625 e^{\frac{26.01}{50}}} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.4

Divide $26.01$ by $50$ .

$f'' (5.1) = - \frac{26.01}{625 e^{0.5202}} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.5

Replace $e$ with an approximation.

$f'' (5.1) = - \frac{26.01}{625 \cdot {2.71828182}^{0.5202}} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.6

Raise $2.71828182$ to the power of $0.5202$ .

$f'' (5.1) = - \frac{26.01}{625 \cdot 1.68236408} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.7

Multiply $625$ by $1.68236408$ .

$f'' (5.1) = - \frac{26.01}{1051.47755554} + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.8

Divide $26.01$ by $1051.47755554$ .

$f'' (5.1) = - 1 \cdot 0.02473661 + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.9

Multiply $- 1$ by $0.02473661$ .

$f'' (5.1) = - 0.02473661 + \frac{e^{- \frac{{(5.1)}^{2}}{50}}}{25}$

Step 7.2.1.10

Move $e^{- \frac{{(5.1)}^{2}}{50}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (5.1) = - 0.02473661 + \frac{1}{25 e^{\frac{{5.1}^{2}}{50}}}$

Step 7.2.1.11

Raise $5.1$ to the power of $2$ .

$f'' (5.1) = - 0.02473661 + \frac{1}{25 e^{\frac{26.01}{50}}}$

Step 7.2.1.12

Divide $26.01$ by $50$ .

$f'' (5.1) = - 0.02473661 + \frac{1}{25 e^{0.5202}}$

Step 7.2.2

Add $- 0.02473661$ and $\frac{1}{25 e^{0.5202}}$ .

$f'' (5.1) = - 0.00096055$

Step 7.2.3

The final answer is $- 0.00096055$ .

$- 0.00096055$

Step 7.3

At $5.1$ , the second derivative is $- 0.00096055$ . Since this is negative, the second derivative is decreasing on the interval $(5, \infty)$

Decreasing on $(5, \infty)$ since $f'' (x) < 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 5, - \frac{1}{e^{\frac{1}{2}}}), (5, - \frac{1}{e^{\frac{1}{2}}})$ .

$(- 5, - \frac{1}{e^{\frac{1}{2}}}), (5, - \frac{1}{e^{\frac{1}{2}}})$

Step 9