Calculus Examples

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

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

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

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

Step 1.1.1.7

Simplify the expression.

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Step 1.1.1.7.1

Add $1$ and $1$ .

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

Step 1.1.1.7.2

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

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

Step 1.1.1.8

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

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

Step 1.1.1.9

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

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

Step 1.1.1.10

Simplify.

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Step 1.1.1.10.1

Reorder terms.

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

Step 1.1.1.10.2

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply by $1$ .

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

Step 1.2.2.3

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

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

Step 1.2.3

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

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

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

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

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

Step 1.2.4.2.2

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

Undefined

Step 1.2.4.2.3

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

No solution

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

Subtract $1$ from both sides of the equation.

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

Step 1.2.5.2.2

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

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Step 1.2.5.2.2.1

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

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

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.2.1.2

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

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

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.2.3

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 1.2.5.2.4

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

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Step 1.2.5.2.4.1

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

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

Step 1.2.5.2.4.2

Any root of $1$ is $1$ .

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

Step 1.2.5.2.4.3

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

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

Step 1.2.5.2.4.4

Combine and simplify the denominator.

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Step 1.2.5.2.4.4.1

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

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

Step 1.2.5.2.4.4.2

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

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

Step 1.2.5.2.4.4.3

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

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

Step 1.2.5.2.4.4.4

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

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

Step 1.2.5.2.4.4.5

Add $1$ and $1$ .

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

Step 1.2.5.2.4.4.6

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

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Step 1.2.5.2.4.4.6.1

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 1.2.5.2.4.4.6.2

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 1.2.5.2.4.4.6.3

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

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

Step 1.2.5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.2.4.4.6.4.1

Cancel the common factor.

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

Step 1.2.5.2.4.4.6.4.2

Rewrite the expression.

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

Step 1.2.5.2.4.4.6.5

Evaluate the exponent.

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

Step 1.2.5.2.5

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

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Step 1.2.5.2.5.1

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

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

Step 1.2.5.2.5.2

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

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

Step 1.2.5.2.5.3

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 1.2.6

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

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

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $x e^{- x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{\sqrt{2}}{2}$ .

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Step 1.4.1.1

Substitute $\frac{\sqrt{2}}{2}$ for $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

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

$\frac{\sqrt{2}}{2} \cdot e^{- \frac{{\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.1.2.2

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

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Step 1.4.1.2.2.1

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

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

Step 1.4.1.2.2.2

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

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

Step 1.4.1.2.2.3

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

$\frac{\sqrt{2}}{2} \cdot e^{- \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.1.2.2.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.2.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot e^{- \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.1.2.2.4.2

Rewrite the expression.

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

Step 1.4.1.2.2.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2} \cdot e^{- \frac{2}{2^{2}}}$

Step 1.4.1.2.3

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

$\frac{\sqrt{2}}{2} \cdot e^{- \frac{2}{4}}$

Step 1.4.1.2.4

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

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Step 1.4.1.2.4.1

Factor $2$ out of $2$ .

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

Step 1.4.1.2.4.2

Cancel the common factors.

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Step 1.4.1.2.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.1.2.4.2.2

Cancel the common factor.

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

Step 1.4.1.2.4.2.3

Rewrite the expression.

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

Step 1.4.1.2.5

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

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

Step 1.4.1.2.6

Combine.

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

Step 1.4.1.2.7

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

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

Step 1.4.2

Evaluate at $x = - \frac{\sqrt{2}}{2}$ .

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Step 1.4.2.1

Substitute $- \frac{\sqrt{2}}{2}$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

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

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Step 1.4.2.2.1.1

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

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

Step 1.4.2.2.1.2

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

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

Step 1.4.2.2.2

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

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Step 1.4.2.2.2.1

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

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

Step 1.4.2.2.2.2

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

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Step 1.4.2.2.2.2.1

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

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

Step 1.4.2.2.2.2.2

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

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

Step 1.4.2.2.2.3

Add $2$ and $1$ .

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

Step 1.4.2.2.3

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

$- \frac{\sqrt{2}}{2} \cdot e^{- \frac{{\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.2.2.4

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

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Step 1.4.2.2.4.1

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

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

Step 1.4.2.2.4.2

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

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

Step 1.4.2.2.4.3

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

$- \frac{\sqrt{2}}{2} \cdot e^{- \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.2.2.4.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.4.1

Cancel the common factor.

$- \frac{\sqrt{2}}{2} \cdot e^{- \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.2.2.4.4.2

Rewrite the expression.

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

Step 1.4.2.2.4.5

Evaluate the exponent.

$- \frac{\sqrt{2}}{2} \cdot e^{- \frac{2}{2^{2}}}$

Step 1.4.2.2.5

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

$- \frac{\sqrt{2}}{2} \cdot e^{- \frac{2}{4}}$

Step 1.4.2.2.6

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

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Step 1.4.2.2.6.1

Factor $2$ out of $2$ .

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

Step 1.4.2.2.6.2

Cancel the common factors.

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Step 1.4.2.2.6.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2.6.2.2

Cancel the common factor.

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

Step 1.4.2.2.6.2.3

Rewrite the expression.

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

Step 1.4.2.2.7

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

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

Step 1.4.2.2.8

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

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

Step 1.4.2.2.9

Move $2$ to the left of $e^{\frac{1}{2}}$ .

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

Step 1.4.3

List all of the points.

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

Step 2

Exclude the points that are not on the interval.

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

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 0$ .

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Step 3.1.1

Substitute $0$ for $x$ .

$(0) \cdot e^{- {(0)}^{2}}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

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

$0 \cdot e^{- 0}$

Step 3.1.2.2

Multiply $- 1$ by $0$ .

$0 \cdot e^{0}$

Step 3.1.2.3

Anything raised to $0$ is $1$ .

$0 \cdot 1$

Step 3.1.2.4

Multiply $0$ by $1$ .

$0$

Step 3.2

Evaluate at $x = 2$ .

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Step 3.2.1

Substitute $2$ for $x$ .

$(2) \cdot e^{- {(2)}^{2}}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$2 \cdot e^{- 1 \cdot 4}$

Step 3.2.2.2

Multiply $- 1$ by $4$ .

$2 \cdot e^{- 4}$

Step 3.2.2.3

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

$2 \cdot \frac{1}{e^{4}}$

Step 3.2.2.4

Combine $2$ and $\frac{1}{e^{4}}$ .

$\frac{2}{e^{4}}$

Step 3.3

List all of the points.

$(0, 0), (2, \frac{2}{e^{4}})$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

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

Absolute Minimum: $(0, 0)$

Step 5