Calculus Examples

$f (x) = x e^{- \frac{x^{2}}{98}}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 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 $g (x) = e^{- \frac{x^{2}}{98}}$ .

$x \frac{d}{d x} [e^{- \frac{x^{2}}{98}}] + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 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 $g (x) = - \frac{x^{2}}{98}$ .

Tap for more steps...

Step 1.2.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{98}$ .

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.2.2

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

$x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.2.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{98}$ .

$x (e^{- \frac{x^{2}}{98}} \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

Since $- \frac{1}{98}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{98}$ with respect to $x$ is $- \frac{1}{98} \frac{d}{d x} [x^{2}]$ .

$x (e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.2

Combine fractions.

Tap for more steps...

Step 1.3.2.1

Combine $e^{- \frac{x^{2}}{98}}$ and $\frac{1}{98}$ .

$x (- \frac{e^{- \frac{x^{2}}{98}}}{98} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{98}}}{98}$ .

$- \frac{x e^{- \frac{x^{2}}{98}}}{98} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.3

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

$- \frac{x e^{- \frac{x^{2}}{98}}}{98} (2 x) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.4

Combine fractions.

Tap for more steps...

Step 1.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{x e^{- \frac{x^{2}}{98}}}{98} x + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{98}}}{98}$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{98}})}{98} x + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.3.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{98}})}{98}$ and $x$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{98}}) x}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.4

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

$\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.5

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

$\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.6

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

$\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.7.1

Add $1$ and $1$ .

$\frac{- 2 (x^{2} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7.2

Cancel the common factor of $- 2$ and $98$ .

Tap for more steps...

Step 1.7.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{98}}$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7.2.2

Cancel the common factors.

Tap for more steps...

Step 1.7.2.2.1

Factor $2$ out of $98$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{2 (49)} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7.2.2.2

Cancel the common factor.

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{2 \cdot 49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7.2.2.3

Rewrite the expression.

$\frac{- x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.7.3

Move the negative in front of the fraction.

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 1.8

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

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \cdot 1$

Step 1.9

Multiply $e^{- \frac{x^{2}}{98}}$ by $1$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49}] + \frac{d}{d x} [e^{- \frac{x^{2}}{98}}]$ .

$f'' (x) = \frac{d}{d x} (- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49}) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49}]$ .

Tap for more steps...

Step 2.2.1

Since $- \frac{1}{49}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49}$ with respect to $x$ is $- \frac{1}{49} \frac{d}{d x} [x^{2} e^{- \frac{x^{2}}{98}}]$ .

$f'' (x) = - \frac{1}{49} \cdot \frac{d}{d x} (x^{2} e^{- \frac{x^{2}}{98}}) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

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 $f (x) = x^{2}$ and $g (x) = e^{- \frac{x^{2}}{98}}$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} \frac{d}{d x} (e^{- \frac{x^{2}}{98}}) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

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 $g (x) = - \frac{x^{2}}{98}$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $- \frac{x^{2}}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (- \frac{x^{2}}{98})) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{u_{1}} \frac{d}{d x} (- \frac{x^{2}}{98})) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $- \frac{x^{2}}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} \frac{d}{d x} (- \frac{x^{2}}{98})) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.4

Since $- \frac{1}{98}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{98}$ with respect to $x$ is $- \frac{1}{98} \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \cdot \frac{d}{d x} x^{2})) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.5

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

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \cdot (2 x))) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.6

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

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \cdot (2 x))) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.7

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (- 2 (\frac{1}{98}) \cdot x)) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.8

Combine $- 2$ and $\frac{1}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{- 2}{98} x)) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.9

Combine $\frac{- 2}{98}$ and $x$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{- 2 x}{98})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.10

Cancel the common factor of $- 2$ and $98$ .

Tap for more steps...

Step 2.2.10.1

Factor $2$ out of $- 2 x$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{98})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.10.2

Cancel the common factors.

Tap for more steps...

Step 2.2.10.2.1

Factor $2$ out of $98$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{2 (49)})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.10.2.2

Cancel the common factor.

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{2 \cdot 49})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.10.2.3

Rewrite the expression.

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (\frac{- x}{49})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (e^{- \frac{x^{2}}{98}} (- \frac{x}{49})) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.12

Combine $e^{- \frac{x^{2}}{98}}$ and $\frac{x}{49}$ .

$f'' (x) = - \frac{1}{49} \cdot (x^{2} (- \frac{e^{- \frac{x^{2}}{98}} x}{49}) + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.13

Combine $x^{2}$ and $\frac{e^{- \frac{x^{2}}{98}} x}{49}$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{2} (e^{- \frac{x^{2}}{98}} x)}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.14

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.14.1

Move $x$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x \cdot x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.14.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 2.2.14.2.1

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

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x \cdot x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.14.2.2

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

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{1 + 2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.2.14.3

Add $1$ and $2$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d x} (e^{- \frac{x^{2}}{98}})$

Step 2.3

Evaluate $\frac{d}{d x} [e^{- \frac{x^{2}}{98}}]$ .

Tap for more steps...

Step 2.3.1

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 $g (x) = - \frac{x^{2}}{98}$ .

Tap for more steps...

Step 2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $- \frac{x^{2}}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + \frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (- \frac{x^{2}}{98})$

Step 2.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{u_{2}} \frac{d}{d x} (- \frac{x^{2}}{98})$

Step 2.3.1.3

Replace all occurrences of $u_{2}$ with $- \frac{x^{2}}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} (- \frac{x^{2}}{98})$

Step 2.3.2

Since $- \frac{1}{98}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{98}$ with respect to $x$ is $- \frac{1}{98} \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \cdot \frac{d}{d x} x^{2})$

Step 2.3.3

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

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \cdot (2 x))$

Step 2.3.4

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (- 2 (\frac{1}{98}) \cdot x)$

Step 2.3.5

Combine $- 2$ and $\frac{1}{98}$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{- 2}{98} x)$

Step 2.3.6

Combine $\frac{- 2}{98}$ and $x$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{- 2 x}{98})$

Step 2.3.7

Cancel the common factor of $- 2$ and $98$ .

Tap for more steps...

Step 2.3.7.1

Factor $2$ out of $- 2 x$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{98})$

Step 2.3.7.2

Cancel the common factors.

Tap for more steps...

Step 2.3.7.2.1

Factor $2$ out of $98$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{2 (49)})$

Step 2.3.7.2.2

Cancel the common factor.

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{2 (- x)}{2 \cdot 49})$

Step 2.3.7.2.3

Rewrite the expression.

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (\frac{- x}{49})$

Step 2.3.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) + e^{- \frac{x^{2}}{98}} (- \frac{x}{49})$

Step 2.3.9

Combine $e^{- \frac{x^{2}}{98}}$ and $\frac{x}{49}$ .

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$f'' (x) = - \frac{1}{49} \cdot (- \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49}) - \frac{1}{49} \cdot (e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{1}{49}) \cdot \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} - \frac{1}{49} \cdot (e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.2

Multiply $\frac{1}{49}$ by $1$ .

$f'' (x) = \frac{1}{49} \cdot \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49} - \frac{1}{49} \cdot (e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.3

Multiply $\frac{1}{49}$ by $\frac{x^{3} e^{- \frac{x^{2}}{98}}}{49}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{49 \cdot 49} - \frac{1}{49} \cdot (e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.4

Multiply $49$ by $49$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - \frac{1}{49} \cdot (e^{- \frac{x^{2}}{98}} (2 x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.5

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - 2 (\frac{1}{49}) \cdot (e^{- \frac{x^{2}}{98}} (x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.6

Combine $- 2$ and $\frac{1}{49}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{- 2}{49} \cdot (e^{- \frac{x^{2}}{98}} (x)) - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.7

Combine $e^{- \frac{x^{2}}{98}}$ and $\frac{- 2}{49}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{e^{- \frac{x^{2}}{98}} \cdot - 2}{49} \cdot x - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.8

Combine $\frac{e^{- \frac{x^{2}}{98}} \cdot - 2}{49}$ and $x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{e^{- \frac{x^{2}}{98}} \cdot (- 2 x)}{49} - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.9

Move $- 2$ to the left of $e^{- \frac{x^{2}}{98}}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{- 2 \cdot (e^{- \frac{x^{2}}{98}} x)}{49} - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.10

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - \frac{2 e^{- \frac{x^{2}}{98}} x}{49} - \frac{e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.11

Combine the numerators over the common denominator.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{- 2 e^{- \frac{x^{2}}{98}} x - e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.12

Subtract $e^{- \frac{x^{2}}{98}} x$ from $- 2 e^{- \frac{x^{2}}{98}} x$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} + \frac{- 3 e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.2.13

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - \frac{3 e^{- \frac{x^{2}}{98}} x}{49}$

Step 2.4.3

Reorder factors in $\frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - \frac{3 e^{- \frac{x^{2}}{98}} x}{49}$ .

$f'' (x) = \frac{x^{3} e^{- \frac{x^{2}}{98}}}{2401} - \frac{3 x e^{- \frac{x^{2}}{98}}}{49}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

$x \frac{d}{d x} [e^{- \frac{x^{2}}{98}}] + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.2

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 $g (x) = - \frac{x^{2}}{98}$ .

Tap for more steps...

Step 4.1.2.1

To apply the Chain Rule, set $u$ as $- \frac{x^{2}}{98}$ .

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.2.2

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

$x (e^{u} \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.2.3

Replace all occurrences of $u$ with $- \frac{x^{2}}{98}$ .

$x (e^{- \frac{x^{2}}{98}} \frac{d}{d x} [- \frac{x^{2}}{98}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3

Differentiate.

Tap for more steps...

Step 4.1.3.1

Since $- \frac{1}{98}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{98}$ with respect to $x$ is $- \frac{1}{98} \frac{d}{d x} [x^{2}]$ .

$x (e^{- \frac{x^{2}}{98}} (- \frac{1}{98} \frac{d}{d x} [x^{2}])) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.2

Combine fractions.

Tap for more steps...

Step 4.1.3.2.1

Combine $e^{- \frac{x^{2}}{98}}$ and $\frac{1}{98}$ .

$x (- \frac{e^{- \frac{x^{2}}{98}}}{98} \frac{d}{d x} [x^{2}]) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.2.2

Combine $x$ and $\frac{e^{- \frac{x^{2}}{98}}}{98}$ .

$- \frac{x e^{- \frac{x^{2}}{98}}}{98} \frac{d}{d x} [x^{2}] + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.3

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

$- \frac{x e^{- \frac{x^{2}}{98}}}{98} (2 x) + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.4

Combine fractions.

Tap for more steps...

Step 4.1.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{x e^{- \frac{x^{2}}{98}}}{98} x + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.4.2

Combine $- 2$ and $\frac{x e^{- \frac{x^{2}}{98}}}{98}$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{98}})}{98} x + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.3.4.3

Combine $\frac{- 2 (x e^{- \frac{x^{2}}{98}})}{98}$ and $x$ .

$\frac{- 2 (x e^{- \frac{x^{2}}{98}}) x}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.4

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

$\frac{- 2 (x^{1} x e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.5

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

$\frac{- 2 (x^{1} x^{1} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.6

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

$\frac{- 2 (x^{1 + 1} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1.7.1

Add $1$ and $1$ .

$\frac{- 2 (x^{2} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7.2

Cancel the common factor of $- 2$ and $98$ .

Tap for more steps...

Step 4.1.7.2.1

Factor $2$ out of $- 2 x^{2} e^{- \frac{x^{2}}{98}}$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{98} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7.2.2

Cancel the common factors.

Tap for more steps...

Step 4.1.7.2.2.1

Factor $2$ out of $98$ .

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{2 (49)} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7.2.2.2

Cancel the common factor.

$\frac{2 (- x^{2} e^{- \frac{x^{2}}{98}})}{2 \cdot 49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7.2.2.3

Rewrite the expression.

$\frac{- x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.7.3

Move the negative in front of the fraction.

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \frac{d}{d x} [x]$

Step 4.1.8

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

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} \cdot 1$

Step 4.1.9

Multiply $e^{- \frac{x^{2}}{98}}$ by $1$ .

$f' (x) = - \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}} = 0$

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

Factor $e^{- \frac{x^{2}}{98}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49} + e^{- \frac{x^{2}}{98}}$ .

Tap for more steps...

Step 5.2.1.1

Factor $e^{- \frac{x^{2}}{98}}$ out of $- \frac{x^{2} e^{- \frac{x^{2}}{98}}}{49}$ .

$e^{- \frac{x^{2}}{98}} (- \frac{x^{2}}{49}) + e^{- \frac{x^{2}}{98}} = 0$

Step 5.2.1.2

Multiply by $1$ .

$e^{- \frac{x^{2}}{98}} (- \frac{x^{2}}{49}) + e^{- \frac{x^{2}}{98}} \cdot 1 = 0$

Step 5.2.1.3

Factor $e^{- \frac{x^{2}}{98}}$ out of $e^{- \frac{x^{2}}{98}} (- \frac{x^{2}}{49}) + e^{- \frac{x^{2}}{98}} \cdot 1$ .

$e^{- \frac{x^{2}}{98}} (- \frac{x^{2}}{49} + 1) = 0$

Step 5.2.2

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

$e^{- \frac{x^{2}}{98}} (- \frac{x^{2}}{49} + 1^{2}) = 0$

Step 5.2.3

Rewrite $\frac{x^{2}}{49}$ as ${(\frac{x}{7})}^{2}$ .

$e^{- \frac{x^{2}}{98}} (- {(\frac{x}{7})}^{2} + 1^{2}) = 0$

Step 5.2.4

Reorder $- {(\frac{x}{7})}^{2}$ and $1^{2}$ .

$e^{- \frac{x^{2}}{98}} (1^{2} - {(\frac{x}{7})}^{2}) = 0$

Step 5.2.5

Factor.

Tap for more steps...

Step 5.2.5.1

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 $b = \frac{x}{7}$ .

$e^{- \frac{x^{2}}{98}} ((1 + \frac{x}{7}) (1 - \frac{x}{7})) = 0$

Step 5.2.5.2

Remove unnecessary parentheses.

$e^{- \frac{x^{2}}{98}} (1 + \frac{x}{7}) (1 - \frac{x}{7}) = 0$

Step 5.3

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

$e^{- \frac{x^{2}}{98}} = 0$

$1 + \frac{x}{7} = 0$

$1 - \frac{x}{7} = 0$

Step 5.4

Set $e^{- \frac{x^{2}}{98}}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.4.1

Set $e^{- \frac{x^{2}}{98}}$ equal to $0$ .

$e^{- \frac{x^{2}}{98}} = 0$

Step 5.4.2

Solve $e^{- \frac{x^{2}}{98}} = 0$ for $x$ .

Tap for more steps...

Step 5.4.2.1

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

$\ln (e^{- \frac{x^{2}}{98}}) = \ln (0)$

Step 5.4.2.2

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

Undefined

Step 5.4.2.3

There is no solution for $e^{- \frac{x^{2}}{98}} = 0$

No solution

Step 5.5

Set $1 + \frac{x}{7}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set $1 + \frac{x}{7}$ equal to $0$ .

$1 + \frac{x}{7} = 0$

Step 5.5.2

Solve $1 + \frac{x}{7} = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

Subtract $1$ from both sides of the equation.

$\frac{x}{7} = - 1$

Step 5.5.2.2

Multiply both sides of the equation by $7$ .

$7 \frac{x}{7} = 7 \cdot - 1$

Step 5.5.2.3

Simplify both sides of the equation.

Tap for more steps...

Step 5.5.2.3.1

Simplify the left side.

Tap for more steps...

Step 5.5.2.3.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.5.2.3.1.1.1

Cancel the common factor.

$7 \frac{x}{7} = 7 \cdot - 1$

Step 5.5.2.3.1.1.2

Rewrite the expression.

$x = 7 \cdot - 1$

Step 5.5.2.3.2

Simplify the right side.

Tap for more steps...

Step 5.5.2.3.2.1

Multiply $7$ by $- 1$ .

$x = - 7$

Step 5.6

Set $1 - \frac{x}{7}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.6.1

Set $1 - \frac{x}{7}$ equal to $0$ .

$1 - \frac{x}{7} = 0$

Step 5.6.2

Solve $1 - \frac{x}{7} = 0$ for $x$ .

Tap for more steps...

Step 5.6.2.1

Subtract $1$ from both sides of the equation.

$- \frac{x}{7} = - 1$

Step 5.6.2.2

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{x}{7}) = - 7 \cdot - 1$

Step 5.6.2.3

Simplify both sides of the equation.

Tap for more steps...

Step 5.6.2.3.1

Simplify the left side.

Tap for more steps...

Step 5.6.2.3.1.1

Simplify $- 7 (- \frac{x}{7})$ .

Tap for more steps...

Step 5.6.2.3.1.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.6.2.3.1.1.1.1

Move the leading negative in $- \frac{x}{7}$ into the numerator.

$- 7 \frac{- x}{7} = - 7 \cdot - 1$

Step 5.6.2.3.1.1.1.2

Factor $7$ out of $- 7$ .

$7 (- 1) \frac{- x}{7} = - 7 \cdot - 1$

Step 5.6.2.3.1.1.1.3

Cancel the common factor.

$7 \cdot - 1 \frac{- x}{7} = - 7 \cdot - 1$

Step 5.6.2.3.1.1.1.4

Rewrite the expression.

$- - x = - 7 \cdot - 1$

Step 5.6.2.3.1.1.2

Multiply.

Tap for more steps...

Step 5.6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 7 \cdot - 1$

Step 5.6.2.3.1.1.2.2

Multiply $x$ by $1$ .

$x = - 7 \cdot - 1$

Step 5.6.2.3.2

Simplify the right side.

Tap for more steps...

Step 5.6.2.3.2.1

Multiply $- 7$ by $- 1$ .

$x = 7$

Step 5.7

The final solution is all the values that make $e^{- \frac{x^{2}}{98}} (1 + \frac{x}{7}) (1 - \frac{x}{7}) = 0$ true.

$x = - 7, 7$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

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 7

Critical points to evaluate.

$x = - 7, 7$

Step 8

Evaluate the second derivative at $x = - 7$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(- 7)}^{3} e^{- \frac{{(- 7)}^{2}}{98}}}{2401} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Move $e^{- \frac{{(- 7)}^{2}}{98}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(- 7)}^{3}}{2401 e^{\frac{{(- 7)}^{2}}{98}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.2

Raise $- 7$ to the power of $2$ .

$\frac{{(- 7)}^{3}}{2401 e^{\frac{49}{98}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.3

Raise $- 7$ to the power of $3$ .

$\frac{- 343}{2401 e^{\frac{49}{98}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.4

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 9.1.4.1

Factor $49$ out of $49$ .

$\frac{- 343}{2401 e^{\frac{49 (1)}{98}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.4.2

Cancel the common factors.

Tap for more steps...

Step 9.1.4.2.1

Factor $49$ out of $98$ .

$\frac{- 343}{2401 e^{\frac{49 \cdot 1}{49 \cdot 2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.4.2.2

Cancel the common factor.

$\frac{- 343}{2401 e^{\frac{49 \cdot 1}{49 \cdot 2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.4.2.3

Rewrite the expression.

$\frac{- 343}{2401 e^{\frac{1}{2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.5

Factor $343$ out of $- 343$ .

$\frac{343 (- 1)}{2401 e^{\frac{1}{2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.6

Cancel the common factors.

Tap for more steps...

Step 9.1.6.1

Factor $343$ out of $2401 e^{\frac{1}{2}}$ .

$\frac{343 (- 1)}{343 (7 e^{\frac{1}{2}})} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.6.2

Cancel the common factor.

$\frac{343 \cdot - 1}{343 (7 e^{\frac{1}{2}})} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.6.3

Rewrite the expression.

$\frac{- 1}{7 e^{\frac{1}{2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.7

Move the negative in front of the fraction.

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{3 (- 7) e^{- \frac{{(- 7)}^{2}}{98}}}{49}$

Step 9.1.8

Move $e^{- \frac{{(- 7)}^{2}}{98}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{3 (- 7)}{49 e^{\frac{{(- 7)}^{2}}{98}}}$

Step 9.1.9

Factor $7$ out of $3 (- 7)$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{7 (3 \cdot (- 1))}{49 e^{\frac{{(- 7)}^{2}}{98}}}$

Step 9.1.10

Cancel the common factors.

Tap for more steps...

Step 9.1.10.1

Factor $7$ out of $49 e^{\frac{{(- 7)}^{2}}{98}}$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{7 (3 \cdot (- 1))}{7 (7 e^{\frac{{(- 7)}^{2}}{98}})}$

Step 9.1.10.2

Cancel the common factor.

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{7 (3 \cdot (- 1))}{7 (7 e^{\frac{{(- 7)}^{2}}{98}})}$

Step 9.1.10.3

Rewrite the expression.

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{3 \cdot (- 1)}{7 e^{\frac{{(- 7)}^{2}}{98}}}$

Step 9.1.11

Raise $- 7$ to the power of $2$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{3 \cdot (- 1)}{7 e^{\frac{49}{98}}}$

Step 9.1.12

Multiply $3$ by $- 1$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{- 3}{7 e^{\frac{49}{98}}}$

Step 9.1.13

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 9.1.13.1

Factor $49$ out of $49$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{- 3}{7 e^{\frac{49 (1)}{98}}}$

Step 9.1.13.2

Cancel the common factors.

Tap for more steps...

Step 9.1.13.2.1

Factor $49$ out of $98$ .

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{- 3}{7 e^{\frac{49 \cdot 1}{49 \cdot 2}}}$

Step 9.1.13.2.2

Cancel the common factor.

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{- 3}{7 e^{\frac{49 \cdot 1}{49 \cdot 2}}}$

Step 9.1.13.2.3

Rewrite the expression.

$- \frac{1}{7 e^{\frac{1}{2}}} - \frac{- 3}{7 e^{\frac{1}{2}}}$

Step 9.1.14

Move the negative in front of the fraction.

$- \frac{1}{7 e^{\frac{1}{2}}} - - \frac{3}{7 e^{\frac{1}{2}}}$

Step 9.1.15

Multiply $- - \frac{3}{7 e^{\frac{1}{2}}}$ .

Tap for more steps...

Step 9.1.15.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{7 e^{\frac{1}{2}}} + 1 \frac{3}{7 e^{\frac{1}{2}}}$

Step 9.1.15.2

Multiply $\frac{3}{7 e^{\frac{1}{2}}}$ by $1$ .

$- \frac{1}{7 e^{\frac{1}{2}}} + \frac{3}{7 e^{\frac{1}{2}}}$

Step 9.2

Combine fractions.

Tap for more steps...

Step 9.2.1

Combine the numerators over the common denominator.

$\frac{- 1 + 3}{7 e^{\frac{1}{2}}}$

Step 9.2.2

Add $- 1$ and $3$ .

$\frac{2}{7 e^{\frac{1}{2}}}$

Step 10

$x = - 7$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 7$ is a local minimum

Step 11

Find the y-value when $x = - 7$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = (- 7) \cdot e^{- \frac{{(- 7)}^{2}}{98}}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Raise $- 7$ to the power of $2$ .

$f (- 7) = - 7 \cdot e^{- \frac{49}{98}}$

Step 11.2.2

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 11.2.2.1

Factor $49$ out of $49$ .

$f (- 7) = - 7 \cdot e^{- \frac{49 (1)}{98}}$

Step 11.2.2.2

Cancel the common factors.

Tap for more steps...

Step 11.2.2.2.1

Factor $49$ out of $98$ .

$f (- 7) = - 7 \cdot e^{- \frac{49 \cdot 1}{49 \cdot 2}}$

Step 11.2.2.2.2

Cancel the common factor.

$f (- 7) = - 7 \cdot e^{- \frac{49 \cdot 1}{49 \cdot 2}}$

Step 11.2.2.2.3

Rewrite the expression.

$f (- 7) = - 7 \cdot e^{- \frac{1}{2}}$

Step 11.2.3

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

$f (- 7) = - 7 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 11.2.4

Combine $- 7$ and $\frac{1}{e^{\frac{1}{2}}}$ .

$f (- 7) = \frac{- 7}{e^{\frac{1}{2}}}$

Step 11.2.5

Move the negative in front of the fraction.

$f (- 7) = - \frac{7}{e^{\frac{1}{2}}}$

Step 11.2.6

The final answer is $- \frac{7}{e^{\frac{1}{2}}}$ .

$y = - \frac{7}{e^{\frac{1}{2}}}$

Step 12

Evaluate the second derivative at $x = 7$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{{(7)}^{3} e^{- \frac{{(7)}^{2}}{98}}}{2401} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Move $e^{- \frac{{(7)}^{2}}{98}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(7)}^{3}}{2401 e^{\frac{7^{2}}{98}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.2

Raise $7$ to the power of $2$ .

$\frac{7^{3}}{2401 e^{\frac{49}{98}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.3

Raise $7$ to the power of $3$ .

$\frac{343}{2401 e^{\frac{49}{98}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.4

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 13.1.4.1

Factor $49$ out of $49$ .

$\frac{343}{2401 e^{\frac{49 (1)}{98}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.4.2

Cancel the common factors.

Tap for more steps...

Step 13.1.4.2.1

Factor $49$ out of $98$ .

$\frac{343}{2401 e^{\frac{49 \cdot 1}{49 \cdot 2}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.4.2.2

Cancel the common factor.

$\frac{343}{2401 e^{\frac{49 \cdot 1}{49 \cdot 2}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.4.2.3

Rewrite the expression.

$\frac{343}{2401 e^{\frac{1}{2}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.5

Factor $343$ out of $343$ .

$\frac{343 (1)}{2401 e^{\frac{1}{2}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.6

Cancel the common factors.

Tap for more steps...

Step 13.1.6.1

Factor $343$ out of $2401 e^{\frac{1}{2}}$ .

$\frac{343 (1)}{343 (7 e^{\frac{1}{2}})} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.6.2

Cancel the common factor.

$\frac{343 \cdot 1}{343 (7 e^{\frac{1}{2}})} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.6.3

Rewrite the expression.

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3 (7) e^{- \frac{{(7)}^{2}}{98}}}{49}$

Step 13.1.7

Move $e^{- \frac{{(7)}^{2}}{98}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3 (7)}{49 e^{\frac{7^{2}}{98}}}$

Step 13.1.8

Factor $7$ out of $3 (7)$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{7 \cdot 3}{49 e^{\frac{7^{2}}{98}}}$

Step 13.1.9

Cancel the common factors.

Tap for more steps...

Step 13.1.9.1

Factor $7$ out of $49 e^{\frac{7^{2}}{98}}$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{7 \cdot 3}{7 (7 e^{\frac{7^{2}}{98}})}$

Step 13.1.9.2

Cancel the common factor.

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{7 \cdot 3}{7 (7 e^{\frac{7^{2}}{98}})}$

Step 13.1.9.3

Rewrite the expression.

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{7^{2}}{98}}}$

Step 13.1.10

Raise $7$ to the power of $2$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{49}{98}}}$

Step 13.1.11

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 13.1.11.1

Factor $49$ out of $49$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{49 (1)}{98}}}$

Step 13.1.11.2

Cancel the common factors.

Tap for more steps...

Step 13.1.11.2.1

Factor $49$ out of $98$ .

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{49 \cdot 1}{49 \cdot 2}}}$

Step 13.1.11.2.2

Cancel the common factor.

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{49 \cdot 1}{49 \cdot 2}}}$

Step 13.1.11.2.3

Rewrite the expression.

$\frac{1}{7 e^{\frac{1}{2}}} - \frac{3}{7 e^{\frac{1}{2}}}$

Step 13.2

Combine fractions.

Tap for more steps...

Step 13.2.1

Combine the numerators over the common denominator.

$\frac{1 - 3}{7 e^{\frac{1}{2}}}$

Step 13.2.2

Simplify the expression.

Tap for more steps...

Step 13.2.2.1

Subtract $3$ from $1$ .

$\frac{- 2}{7 e^{\frac{1}{2}}}$

Step 13.2.2.2

Move the negative in front of the fraction.

$- \frac{2}{7 e^{\frac{1}{2}}}$

Step 14

$x = 7$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 7$ is a local maximum

Step 15

Find the y-value when $x = 7$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = (7) \cdot e^{- \frac{{(7)}^{2}}{98}}$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Raise $7$ to the power of $2$ .

$f (7) = 7 \cdot e^{- \frac{49}{98}}$

Step 15.2.2

Cancel the common factor of $49$ and $98$ .

Tap for more steps...

Step 15.2.2.1

Factor $49$ out of $49$ .

$f (7) = 7 \cdot e^{- \frac{49 (1)}{98}}$

Step 15.2.2.2

Cancel the common factors.

Tap for more steps...

Step 15.2.2.2.1

Factor $49$ out of $98$ .

$f (7) = 7 \cdot e^{- \frac{49 \cdot 1}{49 \cdot 2}}$

Step 15.2.2.2.2

Cancel the common factor.

$f (7) = 7 \cdot e^{- \frac{49 \cdot 1}{49 \cdot 2}}$

Step 15.2.2.2.3

Rewrite the expression.

$f (7) = 7 \cdot e^{- \frac{1}{2}}$

Step 15.2.3

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

$f (7) = 7 \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 15.2.4

Combine $7$ and $\frac{1}{e^{\frac{1}{2}}}$ .

$f (7) = \frac{7}{e^{\frac{1}{2}}}$

Step 15.2.5

The final answer is $\frac{7}{e^{\frac{1}{2}}}$ .

$y = \frac{7}{e^{\frac{1}{2}}}$

Step 16

These are the local extrema for $f (x) = x e^{- \frac{x^{2}}{98}}$ .

$(- 7, - \frac{7}{e^{\frac{1}{2}}})$ is a local minima

$(7, \frac{7}{e^{\frac{1}{2}}})$ is a local maxima

Step 17