Calculus Examples

$f (x) = x^{4} e^{x}$

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 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 $f (x) = x^{4}$ and $g (x) = e^{x}$ .

$f' (x) = x^{4} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{4})$

Step 1.1.2

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

$f' (x) = x^{4} e^{x} + e^{x} \frac{d}{d x} (x^{4})$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = x^{4} e^{x} + e^{x} (4 x^{3})$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

Reorder terms.

$f' (x) = e^{x} x^{4} + 4 e^{x} x^{3}$

Step 1.1.4.2

Reorder factors in $e^{x} x^{4} + 4 e^{x} x^{3}$ .

$f' (x) = x^{4} e^{x} + 4 x^{3} e^{x}$

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $x^{4} e^{x} + 4 x^{3} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{4} e^{x}] + \frac{d}{d x} [4 x^{3} e^{x}]$ .

$f'' (x) = \frac{d}{d x} (x^{4} e^{x}) + \frac{d}{d x} (4 x^{3} e^{x})$

Step 1.2.2

Evaluate $\frac{d}{d x} [x^{4} e^{x}]$ .

Tap for more steps...

Step 1.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{4}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{4} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{4}) + \frac{d}{d x} (4 x^{3} e^{x})$

Step 1.2.2.2

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

$f'' (x) = x^{4} e^{x} + e^{x} \frac{d}{d x} (x^{4}) + \frac{d}{d x} (4 x^{3} e^{x})$

Step 1.2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + \frac{d}{d x} (4 x^{3} e^{x})$

Step 1.2.3

Evaluate $\frac{d}{d x} [4 x^{3} e^{x}]$ .

Tap for more steps...

Step 1.2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3} e^{x}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3} e^{x}]$ .

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 \frac{d}{d x} (x^{3} e^{x})$

Step 1.2.3.2

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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 (x^{3} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{3}))$

Step 1.2.3.3

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

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 (x^{3} e^{x} + e^{x} \frac{d}{d x} (x^{3}))$

Step 1.2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 (x^{3} e^{x} + e^{x} (3 x^{2}))$

Step 1.2.4

Simplify.

Tap for more steps...

Step 1.2.4.1

Apply the distributive property.

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 (x^{3} e^{x}) + 4 (e^{x} (3 x^{2}))$

Step 1.2.4.2

Combine terms.

Tap for more steps...

Step 1.2.4.2.1

Multiply $3$ by $4$ .

$f'' (x) = x^{4} e^{x} + e^{x} (4 x^{3}) + 4 x^{3} e^{x} + 12 (e^{x} (x^{2}))$

Step 1.2.4.2.2

Add $e^{x} (4 x^{3})$ and $4 x^{3} e^{x}$ .

Tap for more steps...

Step 1.2.4.2.2.1

Move $e^{x}$ .

$f'' (x) = x^{4} e^{x} + 4 x^{3} e^{x} + 4 x^{3} e^{x} + 12 e^{x} x^{2}$

Step 1.2.4.2.2.2

Add $4 x^{3} e^{x}$ and $4 x^{3} e^{x}$ .

$f'' (x) = x^{4} e^{x} + 8 x^{3} e^{x} + 12 e^{x} x^{2}$

Step 1.2.4.3

Reorder terms.

$f'' (x) = e^{x} x^{4} + 8 e^{x} x^{3} + 12 e^{x} x^{2}$

Step 1.2.4.4

Reorder factors in $e^{x} x^{4} + 8 e^{x} x^{3} + 12 e^{x} x^{2}$ .

$f'' (x) = x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x}$ .

$x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x}$

Step 2

Set the second derivative equal to $0$ then solve the equation $x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x} = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x} = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor $x^{2} e^{x}$ out of $x^{4} e^{x} + 8 x^{3} e^{x} + 12 x^{2} e^{x}$ .

Tap for more steps...

Step 2.2.1.1

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

$x^{2} e^{x} (x^{2}) + 8 x^{3} e^{x} + 12 x^{2} e^{x} = 0$

Step 2.2.1.2

Factor $x^{2} e^{x}$ out of $8 x^{3} e^{x}$ .

$x^{2} e^{x} (x^{2}) + x^{2} e^{x} (8 x) + 12 x^{2} e^{x} = 0$

Step 2.2.1.3

Factor $x^{2} e^{x}$ out of $12 x^{2} e^{x}$ .

$x^{2} e^{x} (x^{2}) + x^{2} e^{x} (8 x) + x^{2} e^{x} (12) = 0$

Step 2.2.1.4

Factor $x^{2} e^{x}$ out of $x^{2} e^{x} (x^{2}) + x^{2} e^{x} (8 x)$ .

$x^{2} e^{x} (x^{2} + 8 x) + x^{2} e^{x} (12) = 0$

Step 2.2.1.5

Factor $x^{2} e^{x}$ out of $x^{2} e^{x} (x^{2} + 8 x) + x^{2} e^{x} (12)$ .

$x^{2} e^{x} (x^{2} + 8 x + 12) = 0$

Step 2.2.2

Factor.

Tap for more steps...

Step 2.2.2.1

Factor $x^{2} + 8 x + 12$ using the AC method.

Tap for more steps...

Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Step 2.2.2.1.2

Write the factored form using these integers.

$x^{2} e^{x} ((x + 2) (x + 6)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$x^{2} e^{x} (x + 2) (x + 6) = 0$

Step 2.3

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

$x^{2} = 0$

$e^{x} = 0$

$x + 2 = 0$

$x + 6 = 0$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$x^{2} = 0$

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$e^{x} = 0$

Step 2.5.2

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

Tap for more steps...

Step 2.5.2.1

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

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

Step 2.5.2.2

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

Undefined

Step 2.5.2.3

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

No solution

Step 2.6

Set $x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.7

Set $x + 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.7.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.7.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.8

The final solution is all the values that make $x^{2} e^{x} (x + 2) (x + 6) = 0$ true.

$x = 0, - 2, - 6$

Step 3

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

Tap for more steps...

Step 3.1

Substitute $0$ in $f (x) = x^{4} e^{x}$ to find the value of $y$ .

Tap for more steps...

Step 3.1.1

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

$f (0) = {(0)}^{4} e^{0}$

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

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

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

Step 3.1.2.2

Anything raised to $0$ is $1$ .

$f (0) = 0 \cdot 1$

Step 3.1.2.3

Multiply $0$ by $1$ .

$f (0) = 0$

Step 3.1.2.4

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = x^{4} e^{x}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $- 2$ in $f (x) = x^{4} e^{x}$ to find the value of $y$ .

Tap for more steps...

Step 3.3.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 3.3.2

Simplify the result.

Tap for more steps...

Step 3.3.2.1

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

$f (- 2) = 16 e^{- 2}$

Step 3.3.2.2

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

$f (- 2) = 16 (\frac{1}{e^{2}})$

Step 3.3.2.3

Combine $16$ and $\frac{1}{e^{2}}$ .

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

Step 3.3.2.4

The final answer is $\frac{16}{e^{2}}$ .

$\frac{16}{e^{2}}$

Step 3.4

The point found by substituting $- 2$ in $f (x) = x^{4} e^{x}$ is $(- 2, \frac{16}{e^{2}})$ . This point can be an inflection point.

$(- 2, \frac{16}{e^{2}})$

Step 3.5

Substitute $- 6$ in $f (x) = x^{4} e^{x}$ to find the value of $y$ .

Tap for more steps...

Step 3.5.1

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = {(- 6)}^{4} e^{- 6}$

Step 3.5.2

Simplify the result.

Tap for more steps...

Step 3.5.2.1

Raise $- 6$ to the power of $4$ .

$f (- 6) = 1296 e^{- 6}$

Step 3.5.2.2

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

$f (- 6) = 1296 (\frac{1}{e^{6}})$

Step 3.5.2.3

Combine $1296$ and $\frac{1}{e^{6}}$ .

$f (- 6) = \frac{1296}{e^{6}}$

Step 3.5.2.4

The final answer is $\frac{1296}{e^{6}}$ .

$\frac{1296}{e^{6}}$

Step 3.6

The point found by substituting $- 6$ in $f (x) = x^{4} e^{x}$ is $(- 6, \frac{1296}{e^{6}})$ . This point can be an inflection point.

$(- 6, \frac{1296}{e^{6}})$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (- 2, \frac{16}{e^{2}}), (- 6, \frac{1296}{e^{6}})$

Step 4

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

$(- \infty, - 6) \cup (- 6, - 2) \cup (- 2, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 6)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 6.1$ in the expression.

$f'' (- 6.1) = {(- 6.1)}^{4} e^{- 6.1} + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $- 6.1$ to the power of $4$ .

$f'' (- 6.1) = 1384.5841 e^{- 6.1} + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.2

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

$f'' (- 6.1) = 1384.5841 (\frac{1}{e^{6.1}}) + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.3

Combine $1384.5841$ and $\frac{1}{e^{6.1}}$ .

$f'' (- 6.1) = \frac{1384.5841}{e^{6.1}} + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.4

Replace $e$ with an approximation.

$f'' (- 6.1) = \frac{1384.5841}{{2.71828182}^{6.1}} + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.5

Raise $2.71828182$ to the power of $6.1$ .

$f'' (- 6.1) = \frac{1384.5841}{445.85777008} + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.6

Divide $1384.5841$ by $445.85777008$ .

$f'' (- 6.1) = 3.10543898 + 8 {(- 6.1)}^{3} e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.7

Raise $- 6.1$ to the power of $3$ .

$f'' (- 6.1) = 3.10543898 + 8 \cdot (- 226.981 e^{- 6.1}) + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.8

Multiply $8$ by $- 226.981$ .

$f'' (- 6.1) = 3.10543898 - 1815.848 e^{- 6.1} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.9

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

$f'' (- 6.1) = 3.10543898 - 1815.848 \frac{1}{e^{6.1}} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.10

Combine $- 1815.848$ and $\frac{1}{e^{6.1}}$ .

$f'' (- 6.1) = 3.10543898 + \frac{- 1815.848}{e^{6.1}} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.11

Move the negative in front of the fraction.

$f'' (- 6.1) = 3.10543898 - \frac{1815.848}{e^{6.1}} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.12

Replace $e$ with an approximation.

$f'' (- 6.1) = 3.10543898 - \frac{1815.848}{{2.71828182}^{6.1}} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.13

Raise $2.71828182$ to the power of $6.1$ .

$f'' (- 6.1) = 3.10543898 - \frac{1815.848}{445.85777008} + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.14

Divide $1815.848$ by $445.85777008$ .

$f'' (- 6.1) = 3.10543898 - 1 \cdot 4.07270686 + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.15

Multiply $- 1$ by $4.07270686$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + 12 {(- 6.1)}^{2} e^{- 6.1}$

Step 5.2.1.16

Raise $- 6.1$ to the power of $2$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + 12 \cdot (37.21 e^{- 6.1})$

Step 5.2.1.17

Multiply $12$ by $37.21$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + 446.52 e^{- 6.1}$

Step 5.2.1.18

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

$f'' (- 6.1) = 3.10543898 - 4.07270686 + 446.52 (\frac{1}{e^{6.1}})$

Step 5.2.1.19

Combine $446.52$ and $\frac{1}{e^{6.1}}$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + \frac{446.52}{e^{6.1}}$

Step 5.2.1.20

Replace $e$ with an approximation.

$f'' (- 6.1) = 3.10543898 - 4.07270686 + \frac{446.52}{{2.71828182}^{6.1}}$

Step 5.2.1.21

Raise $2.71828182$ to the power of $6.1$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + \frac{446.52}{445.85777008}$

Step 5.2.1.22

Divide $446.52$ by $445.85777008$ .

$f'' (- 6.1) = 3.10543898 - 4.07270686 + 1.00148529$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Subtract $4.07270686$ from $3.10543898$ .

$f'' (- 6.1) = - 0.96726787 + 1.00148529$

Step 5.2.2.2

Add $- 0.96726787$ and $1.00148529$ .

$f'' (- 6.1) = 0.03421741$

Step 5.2.3

The final answer is $0.03421741$ .

$0.03421741$

Step 5.3

At $- 6.1$ , the second derivative is $0.03421741$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 6)$ .

Increasing on $(- \infty, - 6)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 6, - 2)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

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

$f'' (- 4) = {(- 4)}^{4} e^{- 4} + 8 {(- 4)}^{3} e^{- 4} + 12 {(- 4)}^{2} e^{- 4}$

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

$f'' (- 4) = 256 e^{- 4} + 8 {(- 4)}^{3} e^{- 4} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.2

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

$f'' (- 4) = 256 (\frac{1}{e^{4}}) + 8 {(- 4)}^{3} e^{- 4} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.3

Combine $256$ and $\frac{1}{e^{4}}$ .

$f'' (- 4) = \frac{256}{e^{4}} + 8 {(- 4)}^{3} e^{- 4} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.4

Raise $- 4$ to the power of $3$ .

$f'' (- 4) = \frac{256}{e^{4}} + 8 \cdot (- 64 e^{- 4}) + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.5

Multiply $8$ by $- 64$ .

$f'' (- 4) = \frac{256}{e^{4}} - 512 e^{- 4} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.6

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

$f'' (- 4) = \frac{256}{e^{4}} - 512 \frac{1}{e^{4}} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.7

Combine $- 512$ and $\frac{1}{e^{4}}$ .

$f'' (- 4) = \frac{256}{e^{4}} + \frac{- 512}{e^{4}} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.8

Move the negative in front of the fraction.

$f'' (- 4) = \frac{256}{e^{4}} - \frac{512}{e^{4}} + 12 {(- 4)}^{2} e^{- 4}$

Step 6.2.1.9

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

$f'' (- 4) = \frac{256}{e^{4}} - \frac{512}{e^{4}} + 12 \cdot (16 e^{- 4})$

Step 6.2.1.10

Multiply $12$ by $16$ .

$f'' (- 4) = \frac{256}{e^{4}} - \frac{512}{e^{4}} + 192 e^{- 4}$

Step 6.2.1.11

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

$f'' (- 4) = \frac{256}{e^{4}} - \frac{512}{e^{4}} + 192 (\frac{1}{e^{4}})$

Step 6.2.1.12

Combine $192$ and $\frac{1}{e^{4}}$ .

$f'' (- 4) = \frac{256}{e^{4}} - \frac{512}{e^{4}} + \frac{192}{e^{4}}$

Step 6.2.2

Combine fractions.

Tap for more steps...

Step 6.2.2.1

Combine the numerators over the common denominator.

$f'' (- 4) = \frac{256 - 512 + 192}{e^{4}}$

Step 6.2.2.2

Simplify the expression.

Tap for more steps...

Step 6.2.2.2.1

Subtract $512$ from $256$ .

$f'' (- 4) = \frac{- 256 + 192}{e^{4}}$

Step 6.2.2.2.2

Add $- 256$ and $192$ .

$f'' (- 4) = \frac{- 64}{e^{4}}$

Step 6.2.2.2.3

Move the negative in front of the fraction.

$f'' (- 4) = - \frac{64}{e^{4}}$

Step 6.2.3

The final answer is $- \frac{64}{e^{4}}$ .

$- \frac{64}{e^{4}}$

Step 6.3

At $- 4$ , the second derivative is $- 1.17220088$ . Since this is negative, the second derivative is decreasing on the interval $(- 6, - 2)$

Decreasing on $(- 6, - 2)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 2, 0)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f'' (- 1) = {(- 1)}^{4} e^{- 1} + 8 {(- 1)}^{3} e^{- 1} + 12 {(- 1)}^{2} e^{- 1}$

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

Raise $- 1$ to the power of $4$ .

$f'' (- 1) = 1 e^{- 1} + 8 {(- 1)}^{3} e^{- 1} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.2

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

$f'' (- 1) = e^{- 1} + 8 {(- 1)}^{3} e^{- 1} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.3

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

$f'' (- 1) = \frac{1}{e} + 8 {(- 1)}^{3} e^{- 1} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.4

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

$f'' (- 1) = \frac{1}{e} + 8 \cdot (- 1 e^{- 1}) + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.5

Multiply $8$ by $- 1$ .

$f'' (- 1) = \frac{1}{e} - 8 e^{- 1} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.6

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

$f'' (- 1) = \frac{1}{e} - 8 \frac{1}{e} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.7

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

$f'' (- 1) = \frac{1}{e} + \frac{- 8}{e} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.8

Move the negative in front of the fraction.

$f'' (- 1) = \frac{1}{e} - \frac{8}{e} + 12 {(- 1)}^{2} e^{- 1}$

Step 7.2.1.9

Raise $- 1$ to the power of $2$ .

$f'' (- 1) = \frac{1}{e} - \frac{8}{e} + 12 \cdot (1 e^{- 1})$

Step 7.2.1.10

Multiply $12$ by $1$ .

$f'' (- 1) = \frac{1}{e} - \frac{8}{e} + 12 e^{- 1}$

Step 7.2.1.11

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

$f'' (- 1) = \frac{1}{e} - \frac{8}{e} + 12 (\frac{1}{e})$

Step 7.2.1.12

Combine $12$ and $\frac{1}{e}$ .

$f'' (- 1) = \frac{1}{e} - \frac{8}{e} + \frac{12}{e}$

Step 7.2.2

Combine fractions.

Tap for more steps...

Step 7.2.2.1

Combine the numerators over the common denominator.

$f'' (- 1) = \frac{1 - 8 + 12}{e}$

Step 7.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 7.2.2.2.1

Subtract $8$ from $1$ .

$f'' (- 1) = \frac{- 7 + 12}{e}$

Step 7.2.2.2.2

Add $- 7$ and $12$ .

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

Step 7.2.3

The final answer is $\frac{5}{e}$ .

$\frac{5}{e}$

Step 7.3

At $- 1$ , the second derivative is $1.8393972$ . Since this is positive, the second derivative is increasing on the interval $(- 2, 0)$ .

Increasing on $(- 2, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = {(0.1)}^{4} e^{0.1} + 8 {(0.1)}^{3} e^{0.1} + 12 {(0.1)}^{2} e^{0.1}$

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 $0.1$ to the power of $4$ .

$f'' (0.1) = 0.0001 e^{0.1} + 8 {(0.1)}^{3} e^{0.1} + 12 {(0.1)}^{2} e^{0.1}$

Step 8.2.1.2

Multiply $0.0001$ by $e^{0.1}$ .

$f'' (0.1) = 0.00011051 + 8 {(0.1)}^{3} e^{0.1} + 12 {(0.1)}^{2} e^{0.1}$

Step 8.2.1.3

Raise $0.1$ to the power of $3$ .

$f'' (0.1) = 0.00011051 + 8 \cdot (0.001 e^{0.1}) + 12 {(0.1)}^{2} e^{0.1}$

Step 8.2.1.4

Multiply $8$ by $0.001$ .

$f'' (0.1) = 0.00011051 + 0.008 e^{0.1} + 12 {(0.1)}^{2} e^{0.1}$

Step 8.2.1.5

Multiply $0.008$ by $e^{0.1}$ .

$f'' (0.1) = 0.00011051 + 0.00884136 + 12 {(0.1)}^{2} e^{0.1}$

Step 8.2.1.6

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = 0.00011051 + 0.00884136 + 12 \cdot (0.01 e^{0.1})$

Step 8.2.1.7

Multiply $12$ by $0.01$ .

$f'' (0.1) = 0.00011051 + 0.00884136 + 0.12 e^{0.1}$

Step 8.2.1.8

Multiply $0.12$ by $e^{0.1}$ .

$f'' (0.1) = 0.00011051 + 0.00884136 + 0.13262051$

Step 8.2.2

Simplify by adding numbers.

Tap for more steps...

Step 8.2.2.1

Add $0.00011051$ and $0.00884136$ .

$f'' (0.1) = 0.00895188 + 0.13262051$

Step 8.2.2.2

Add $0.00895188$ and $0.13262051$ .

$f'' (0.1) = 0.14157239$

Step 8.2.3

The final answer is $0.14157239$ .

$0.14157239$

Step 8.3

At $0.1$ , the second derivative is $0.14157239$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ 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 $(- 6, \frac{1296}{e^{6}}), (- 2, \frac{16}{e^{2}})$ .

$(- 6, \frac{1296}{e^{6}}), (- 2, \frac{16}{e^{2}})$

Step 10