Calculus Examples

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

Step 1

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

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

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

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

Step 2.1.2.2

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

$e^{- x^{2}} (- (2 x))$

Step 2.1.2.3

Multiply $2$ by $- 1$ .

$e^{- x^{2}} (- 2 x)$

Step 2.1.3

Simplify.

Tap for more steps...

Step 2.1.3.1

Reorder the factors of $e^{- x^{2}} (- 2 x)$ .

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

Step 2.1.3.2

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

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

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

Tap for more steps...

Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

Differentiate.

Tap for more steps...

Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify the expression.

Tap for more steps...

Step 2.2.8.1

Add $1$ and $1$ .

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

Step 2.2.8.2

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

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

Step 2.2.9

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

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

Step 2.2.10

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

$- 2 (x^{2} (- 2 e^{- x^{2}}) + e^{- x^{2}})$

Step 2.2.11

Simplify.

Tap for more steps...

Step 2.2.11.1

Apply the distributive property.

$- 2 (x^{2} (- 2 e^{- x^{2}})) - 2 e^{- x^{2}}$

Step 2.2.11.2

Multiply $- 2$ by $- 2$ .

$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$

Step 2.2.11.3

Reorder terms.

$4 e^{- x^{2}} x^{2} - 2 e^{- x^{2}}$

Step 2.2.11.4

Reorder factors in $4 e^{- x^{2}} x^{2} - 2 e^{- x^{2}}$ .

$f'' (x) = 4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$ .

$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}} = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}} = 0$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

$2 x^{2} - 1 = 0$

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

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

Tap for more steps...

Step 3.4.2.1

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

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

Step 3.4.2.2

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

Undefined

Step 3.4.2.3

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

No solution

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$2 x^{2} - 1 = 0$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

Add $1$ to both sides of the equation.

$2 x^{2} = 1$

Step 3.5.2.2

Divide each term in $2 x^{2} = 1$ by $2$ and simplify.

Tap for more steps...

Step 3.5.2.2.1

Divide each term in $2 x^{2} = 1$ by $2$ .

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

Step 3.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 3.5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{2}}$

Step 3.5.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Tap for more steps...

Step 3.5.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.5.2.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 3.5.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.5.2.4.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.5.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.5.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.5.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.5.2.4.4.4

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

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.5.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.5.2.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.5.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.5.2.4.4.6.3

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

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.2.4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 3.5.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

Step 3.5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{2}$

Step 3.5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.5.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 3.6

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

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Replace the variable $x$ with $\frac{\sqrt{2}}{2}$ in the expression.

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

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 4.1.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 4.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 4.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.1.2.2.3

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

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

Step 4.1.2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.2.4.1

Cancel the common factor.

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

Step 4.1.2.2.4.2

Rewrite the expression.

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

Step 4.1.2.2.5

Evaluate the exponent.

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

Step 4.1.2.3

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

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

Step 4.1.2.4

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

Tap for more steps...

Step 4.1.2.4.1

Factor $2$ out of $2$ .

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

Step 4.1.2.4.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.4.2.1

Factor $2$ out of $4$ .

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

Step 4.1.2.4.2.2

Cancel the common factor.

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

Step 4.1.2.4.2.3

Rewrite the expression.

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

Step 4.1.2.5

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

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

Step 4.1.2.6

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

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

Step 4.2

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

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Replace the variable $x$ with $- \frac{\sqrt{2}}{2}$ in the expression.

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

Step 4.3.2

Simplify the result.

Tap for more steps...

Step 4.3.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.3.2.1.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

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

Step 4.3.2.1.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 4.3.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 4.3.2.2.1

Move ${(- 1)}^{2}$ .

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

Step 4.3.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

Tap for more steps...

Step 4.3.2.2.2.1

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

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

Step 4.3.2.2.2.2

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

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

Step 4.3.2.2.3

Add $2$ and $1$ .

$f (- \frac{\sqrt{2}}{2}) = e^{{(- 1)}^{3} (\frac{{\sqrt{2}}^{2}}{2^{2}})}$

Step 4.3.2.3

Raise $- 1$ to the power of $3$ .

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

Step 4.3.2.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 4.3.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 4.3.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2.4.3

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

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

Step 4.3.2.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.4.4.1

Cancel the common factor.

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

Step 4.3.2.4.4.2

Rewrite the expression.

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

Step 4.3.2.4.5

Evaluate the exponent.

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

Step 4.3.2.5

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

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

Step 4.3.2.6

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

Tap for more steps...

Step 4.3.2.6.1

Factor $2$ out of $2$ .

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

Step 4.3.2.6.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.6.2.1

Factor $2$ out of $4$ .

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

Step 4.3.2.6.2.2

Cancel the common factor.

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

Step 4.3.2.6.2.3

Rewrite the expression.

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

Step 4.3.2.7

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

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

Step 4.3.2.8

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

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

Step 4.4

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

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

Step 4.5

Determine the points that could be inflection points.

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

Step 5

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

$(- \infty, - \frac{\sqrt{2}}{2}) \cup (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \cup (\frac{\sqrt{2}}{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 0.80710678$ in the expression.

$f'' (- 0.80710678) = 4 {(- 0.80710678)}^{2} e^{- {(- 0.80710678)}^{2}} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $- 0.80710678$ to the power of $2$ .

$f'' (- 0.80710678) = 4 \cdot (0.65142135 e^{- {(- 0.80710678)}^{2}}) - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.2

Multiply $4$ by $0.65142135$ .

$f'' (- 0.80710678) = 2.60568542 e^{- {(- 0.80710678)}^{2}} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.3

Raise $- 0.80710678$ to the power of $2$ .

$f'' (- 0.80710678) = 2.60568542 e^{- 1 \cdot 0.65142135} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.4

Multiply $- 1$ by $0.65142135$ .

$f'' (- 0.80710678) = 2.60568542 e^{- 0.65142135} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.5

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

$f'' (- 0.80710678) = 2.60568542 (\frac{1}{e^{0.65142135}}) - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.6

Combine $2.60568542$ and $\frac{1}{e^{0.65142135}}$ .

$f'' (- 0.80710678) = \frac{2.60568542}{e^{0.65142135}} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.7

Replace $e$ with an approximation.

$f'' (- 0.80710678) = \frac{2.60568542}{{2.71828182}^{0.65142135}} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.8

Raise $2.71828182$ to the power of $0.65142135$ .

$f'' (- 0.80710678) = \frac{2.60568542}{1.91826543} - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.9

Divide $2.60568542$ by $1.91826543$ .

$f'' (- 0.80710678) = 1.35835499 - 2 e^{- {(- 0.80710678)}^{2}}$

Step 6.2.1.10

Raise $- 0.80710678$ to the power of $2$ .

$f'' (- 0.80710678) = 1.35835499 - 2 e^{- 1 \cdot 0.65142135}$

Step 6.2.1.11

Multiply $- 1$ by $0.65142135$ .

$f'' (- 0.80710678) = 1.35835499 - 2 e^{- 0.65142135}$

Step 6.2.1.12

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

$f'' (- 0.80710678) = 1.35835499 - 2 \frac{1}{e^{0.65142135}}$

Step 6.2.1.13

Combine $- 2$ and $\frac{1}{e^{0.65142135}}$ .

$f'' (- 0.80710678) = 1.35835499 + \frac{- 2}{e^{0.65142135}}$

Step 6.2.1.14

Move the negative in front of the fraction.

$f'' (- 0.80710678) = 1.35835499 - \frac{2}{e^{0.65142135}}$

Step 6.2.2

Subtract $\frac{2}{e^{0.65142135}}$ from $1.35835499$ .

$f'' (- 0.80710678) = 0.31574641$

Step 6.2.3

The final answer is $0.31574641$ .

$0.31574641$

Step 6.3

At $- 0.80710678$ , the second derivative is $0.31574641$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \frac{\sqrt{2}}{2})$ .

Increasing on $(- \infty, - \frac{\sqrt{2}}{2})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

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

$f'' (0) = 4 {(0)}^{2} e^{- {(0)}^{2}} - 2 e^{- {(0)}^{2}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

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

$f'' (0) = 4 \cdot (0 e^{- {(0)}^{2}}) - 2 e^{- {(0)}^{2}}$

Step 7.2.1.2

Multiply $4$ by $0$ .

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Multiply $- 1$ by $0$ .

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

Step 7.2.1.5

Anything raised to $0$ is $1$ .

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

Step 7.2.1.6

Multiply $0$ by $1$ .

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

Step 7.2.1.7

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

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

Step 7.2.1.8

Multiply $- 1$ by $0$ .

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

Step 7.2.1.9

Anything raised to $0$ is $1$ .

$f'' (0) = 0 - 2 \cdot 1$

Step 7.2.1.10

Multiply $- 2$ by $1$ .

$f'' (0) = 0 - 2$

Step 7.2.2

Subtract $2$ from $0$ .

$f'' (0) = - 2$

Step 7.2.3

The final answer is $- 2$ .

$- 2$

Step 7.3

At $0$ , the second derivative is $- 2$ . Since this is negative, the second derivative is decreasing on the interval $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Decreasing on $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(\frac{\sqrt{2}}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $0.80710678$ in the expression.

$f'' (0.80710678) = 4 {(0.80710678)}^{2} e^{- {(0.80710678)}^{2}} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Raise $0.80710678$ to the power of $2$ .

$f'' (0.80710678) = 4 \cdot (0.65142135 e^{- {(0.80710678)}^{2}}) - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.2

Multiply $4$ by $0.65142135$ .

$f'' (0.80710678) = 2.60568542 e^{- {(0.80710678)}^{2}} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.3

Raise $0.80710678$ to the power of $2$ .

$f'' (0.80710678) = 2.60568542 e^{- 1 \cdot 0.65142135} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.4

Multiply $- 1$ by $0.65142135$ .

$f'' (0.80710678) = 2.60568542 e^{- 0.65142135} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.5

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

$f'' (0.80710678) = 2.60568542 (\frac{1}{e^{0.65142135}}) - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.6

Combine $2.60568542$ and $\frac{1}{e^{0.65142135}}$ .

$f'' (0.80710678) = \frac{2.60568542}{e^{0.65142135}} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.7

Replace $e$ with an approximation.

$f'' (0.80710678) = \frac{2.60568542}{{2.71828182}^{0.65142135}} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.8

Raise $2.71828182$ to the power of $0.65142135$ .

$f'' (0.80710678) = \frac{2.60568542}{1.91826543} - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.9

Divide $2.60568542$ by $1.91826543$ .

$f'' (0.80710678) = 1.35835499 - 2 e^{- {(0.80710678)}^{2}}$

Step 8.2.1.10

Raise $0.80710678$ to the power of $2$ .

$f'' (0.80710678) = 1.35835499 - 2 e^{- 1 \cdot 0.65142135}$

Step 8.2.1.11

Multiply $- 1$ by $0.65142135$ .

$f'' (0.80710678) = 1.35835499 - 2 e^{- 0.65142135}$

Step 8.2.1.12

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

$f'' (0.80710678) = 1.35835499 - 2 \frac{1}{e^{0.65142135}}$

Step 8.2.1.13

Combine $- 2$ and $\frac{1}{e^{0.65142135}}$ .

$f'' (0.80710678) = 1.35835499 + \frac{- 2}{e^{0.65142135}}$

Step 8.2.1.14

Move the negative in front of the fraction.

$f'' (0.80710678) = 1.35835499 - \frac{2}{e^{0.65142135}}$

Step 8.2.2

Subtract $\frac{2}{e^{0.65142135}}$ from $1.35835499$ .

$f'' (0.80710678) = 0.31574641$

Step 8.2.3

The final answer is $0.31574641$ .

$0.31574641$

Step 8.3

At $0.80710678$ , the second derivative is $0.31574641$ . Since this is positive, the second derivative is increasing on the interval $(\frac{\sqrt{2}}{2}, \infty)$ .

Increasing on $(\frac{\sqrt{2}}{2}, \infty)$ since $f'' (x) > 0$

Step 9

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

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

Step 10