Calculus Examples

$(x^{2} - 6) e^{x}$

Step 1

Write $(x^{2} - 6) e^{x}$ as a function.

$f (x) = (x^{2} - 6) e^{x}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $f (x) = x^{2} - 6$ and $g (x) = e^{x}$ .

$f' (x) = (x^{2} - 6) \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{2} - 6)$

Step 2.1.2

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

$f' (x) = (x^{2} - 6) e^{x} + e^{x} \frac{d}{d x} (x^{2} - 6)$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

By the Sum Rule, the derivative of $x^{2} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]$ .

$f' (x) = (x^{2} - 6) e^{x} + e^{x} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6))$

Step 2.1.3.2

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

$f' (x) = (x^{2} - 6) e^{x} + e^{x} (2 x + \frac{d}{d x} (- 6))$

Step 2.1.3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

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

Step 2.1.3.4

Add $2 x$ and $0$ .

$f' (x) = (x^{2} - 6) e^{x} + e^{x} (2 x)$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$f' (x) = x^{2} e^{x} - 6 e^{x} + e^{x} (2 x)$

Step 2.1.4.2

Reorder terms.

$f' (x) = e^{x} x^{2} + 2 e^{x} x - 6 e^{x}$

Step 2.1.4.3

Reorder factors in $e^{x} x^{2} + 2 e^{x} x - 6 e^{x}$ .

$f' (x) = x^{2} e^{x} + 2 x e^{x} - 6 e^{x}$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} e^{x} + 2 x e^{x} - 6 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [2 x e^{x}] + \frac{d}{d x} [- 6 e^{x}]$ .

$f'' (x) = \frac{d}{d x} (x^{2} e^{x}) + \frac{d}{d x} (2 x e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.2

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

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Step 2.2.2.1

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

$f'' (x) = x^{2} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{2}) + \frac{d}{d x} (2 x e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.2.2

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

$f'' (x) = x^{2} e^{x} + e^{x} \frac{d}{d x} (x^{2}) + \frac{d}{d x} (2 x e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.2.3

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + \frac{d}{d x} (2 x e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.3

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

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Step 2.2.3.1

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 \frac{d}{d x} (x e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.3.2

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.3.3

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x} + e^{x} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.3.4

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.3.5

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x} + e^{x}) + \frac{d}{d x} (- 6 e^{x})$

Step 2.2.4

Evaluate $\frac{d}{d x} [- 6 e^{x}]$ .

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Step 2.2.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 e^{x}$ with respect to $x$ is $- 6 \frac{d}{d x} [e^{x}]$ .

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x} + e^{x}) - 6 \frac{d}{d x} e^{x}$

Step 2.2.4.2

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

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x} + e^{x}) - 6 e^{x}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Apply the distributive property.

$f'' (x) = x^{2} e^{x} + e^{x} (2 x) + 2 (x e^{x}) + 2 e^{x} - 6 e^{x}$

Step 2.2.5.2

Combine terms.

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Step 2.2.5.2.1

Add $e^{x} (2 x)$ and $2 x e^{x}$ .

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Step 2.2.5.2.1.1

Move $e^{x}$ .

$f'' (x) = x^{2} e^{x} + 2 x e^{x} + 2 x e^{x} + 2 e^{x} - 6 e^{x}$

Step 2.2.5.2.1.2

Add $2 x e^{x}$ and $2 x e^{x}$ .

$f'' (x) = x^{2} e^{x} + 4 x e^{x} + 2 e^{x} - 6 e^{x}$

Step 2.2.5.2.2

Subtract $6 e^{x}$ from $2 e^{x}$ .

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

Step 2.2.5.3

Reorder terms.

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

Step 2.2.5.4

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$e^{x} x^{2} + e^{x} (4 x) - 4 e^{x} = 0$

Step 3.2.3

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

$e^{x} x^{2} + e^{x} (4 x) + e^{x} \cdot - 4 = 0$

Step 3.2.4

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

$e^{x} (x^{2} + 4 x) + e^{x} \cdot - 4 = 0$

Step 3.2.5

Factor $e^{x}$ out of $e^{x} (x^{2} + 4 x) + e^{x} \cdot - 4$ .

$e^{x} (x^{2} + 4 x - 4) = 0$

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 $0$ .

$e^{x} = 0$

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

Step 3.4

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

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Step 3.4.1

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

$e^{x} = 0$

Step 3.4.2

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

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Step 3.4.2.1

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

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

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 $e^{x} = 0$

No solution

Step 3.5

Set $x^{2} + 4 x - 4$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x^{2} + 4 x - 4$ equal to $0$ .

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

Step 3.5.2

Solve $x^{2} + 4 x - 4 = 0$ for $x$ .

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Step 3.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 3.5.2.3

Simplify.

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Step 3.5.2.3.1

Simplify the numerator.

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Step 3.5.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.3.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 3.5.2.3.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 3.5.2.3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.5.2.3.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 3.5.2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 3.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 3.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Step 3.5.2.3.3

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

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

Step 3.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.5.2.4.1

Simplify the numerator.

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Step 3.5.2.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.4.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 3.5.2.4.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 3.5.2.4.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.5.2.4.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 3.5.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 3.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 3.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Step 3.5.2.4.3

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

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

Step 3.5.2.4.4

Change the $\pm$ to $+$ .

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

Step 3.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.5.2.5.1

Simplify the numerator.

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Step 3.5.2.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.5.2.5.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Step 3.5.2.5.1.3

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Step 3.5.2.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.5.2.5.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 3.5.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 3.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 3.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Step 3.5.2.5.3

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

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

Step 3.5.2.5.4

Change the $\pm$ to $-$ .

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

Step 3.5.2.6

The final answer is the combination of both solutions.

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

Step 3.6

The final solution is all the values that make $e^{x} (x^{2} + 4 x - 4) = 0$ true.

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

Step 4

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

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Step 4.1

Substitute $- 2 + 2 \sqrt{2}$ in $f (x) = (x^{2} - 6) e^{x}$ to find the value of $y$ .

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Step 4.1.1

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

$f (- 2 + 2 \sqrt{2}) = ({(- 2 + 2 \sqrt{2})}^{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Rewrite ${(- 2 + 2 \sqrt{2})}^{2}$ as $(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})$ .

$f (- 2 + 2 \sqrt{2}) = ((- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.2

Expand $(- 2 + 2 \sqrt{2}) (- 2 + 2 \sqrt{2})$ using the FOIL Method.

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Step 4.1.2.1.2.1

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = (- 2 (- 2 + 2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.2.2

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = (- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} (- 2 + 2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.2.3

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = (- 2 \cdot - 2 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3

Simplify and combine like terms.

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Step 4.1.2.1.3.1

Simplify each term.

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Step 4.1.2.1.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 2 (2 \sqrt{2}) + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.2

Multiply $2$ by $- 2$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} + 2 \sqrt{2} \cdot - 2 + 2 \sqrt{2} (2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.3

Multiply $- 2$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

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Step 4.1.2.1.3.1.4.1

Multiply $2$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.4.2

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.4.3

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.4.4

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.4.5

Add $1$ and $1$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {\sqrt{2}}^{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5

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

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Step 4.1.2.1.3.1.5.1

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5.2

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5.3

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

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.3.1.5.4.1

Cancel the common factor.

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5.4.2

Rewrite the expression.

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2 - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.5.5

Evaluate the exponent.

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 4 \cdot 2 - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.1.6

Multiply $4$ by $2$ .

$f (- 2 + 2 \sqrt{2}) = (4 - 4 \sqrt{2} - 4 \sqrt{2} + 8 - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.2

Add $4$ and $8$ .

$f (- 2 + 2 \sqrt{2}) = (12 - 4 \sqrt{2} - 4 \sqrt{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.1.3.3

Subtract $4 \sqrt{2}$ from $- 4 \sqrt{2}$ .

$f (- 2 + 2 \sqrt{2}) = (12 - 8 \sqrt{2} - 6) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.2

Simplify by multiplying through.

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Step 4.1.2.2.1

Subtract $6$ from $12$ .

$f (- 2 + 2 \sqrt{2}) = (6 - 8 \sqrt{2}) \cdot e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.2.2

Apply the distributive property.

$f (- 2 + 2 \sqrt{2}) = 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}}$

Step 4.1.2.3

The final answer is $6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}}$ .

$6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}}$

Step 4.2

The point found by substituting $- 2 + 2 \sqrt{2}$ in $f (x) = (x^{2} - 6) e^{x}$ is $(- 2 + 2 \sqrt{2}, 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}})$ . This point can be an inflection point.

$(- 2 + 2 \sqrt{2}, 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}})$

Step 4.3

Substitute $- 2 - 2 \sqrt{2}$ in $f (x) = (x^{2} - 6) e^{x}$ to find the value of $y$ .

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Step 4.3.1

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

$f (- 2 - 2 \sqrt{2}) = ({(- 2 - 2 \sqrt{2})}^{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Rewrite ${(- 2 - 2 \sqrt{2})}^{2}$ as $(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})$ .

$f (- 2 - 2 \sqrt{2}) = ((- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.2

Expand $(- 2 - 2 \sqrt{2}) (- 2 - 2 \sqrt{2})$ using the FOIL Method.

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Step 4.3.2.1.2.1

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = (- 2 (- 2 - 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.2.2

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = (- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} (- 2 - 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.2.3

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = (- 2 \cdot - 2 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3

Simplify and combine like terms.

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Step 4.3.2.1.3.1

Simplify each term.

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Step 4.3.2.1.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = (4 - 2 (- 2 \sqrt{2}) - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.2

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} - 2 \sqrt{2} \cdot - 2 - 2 \sqrt{2} (- 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} - 2 \sqrt{2} (- 2 \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.4

Multiply $- 2 \sqrt{2} (- 2 \sqrt{2})$ .

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Step 4.3.2.1.3.1.4.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \sqrt{2} \sqrt{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.4.2

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.4.3

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 (\sqrt{2} \sqrt{2}) - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.4.4

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.4.5

Add $1$ and $1$ .

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {\sqrt{2}}^{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5

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

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Step 4.3.2.1.3.1.5.1

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5.2

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5.3

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

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.3.1.5.4.1

Cancel the common factor.

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5.4.2

Rewrite the expression.

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2 - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.5.5

Evaluate the exponent.

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 4 \cdot 2 - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.1.6

Multiply $4$ by $2$ .

$f (- 2 - 2 \sqrt{2}) = (4 + 4 \sqrt{2} + 4 \sqrt{2} + 8 - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.2

Add $4$ and $8$ .

$f (- 2 - 2 \sqrt{2}) = (12 + 4 \sqrt{2} + 4 \sqrt{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.1.3.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$f (- 2 - 2 \sqrt{2}) = (12 + 8 \sqrt{2} - 6) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.2

Simplify by multiplying through.

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Step 4.3.2.2.1

Subtract $6$ from $12$ .

$f (- 2 - 2 \sqrt{2}) = (6 + 8 \sqrt{2}) \cdot e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.2.2

Apply the distributive property.

$f (- 2 - 2 \sqrt{2}) = 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}}$

Step 4.3.2.3

The final answer is $6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}}$ .

$6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}}$

Step 4.4

The point found by substituting $- 2 - 2 \sqrt{2}$ in $f (x) = (x^{2} - 6) e^{x}$ is $(- 2 - 2 \sqrt{2}, 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}})$ . This point can be an inflection point.

$(- 2 - 2 \sqrt{2}, 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}})$

Step 4.5

Determine the points that could be inflection points.

$(- 2 + 2 \sqrt{2}, 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}}), (- 2 - 2 \sqrt{2}, 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}})$

Step 5

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

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 4.92842712$ in the expression.

$f'' (- 4.92842712) = {(- 4.92842712)}^{2} e^{- 4.92842712} + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 4.92842712$ to the power of $2$ .

$f'' (- 4.92842712) = 24.28939392 e^{- 4.92842712} + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.2

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

$f'' (- 4.92842712) = 24.28939392 (\frac{1}{e^{4.92842712}}) + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.3

Combine $24.28939392$ and $\frac{1}{e^{4.92842712}}$ .

$f'' (- 4.92842712) = \frac{24.28939392}{e^{4.92842712}} + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.4

Replace $e$ with an approximation.

$f'' (- 4.92842712) = \frac{24.28939392}{{2.71828182}^{4.92842712}} + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.5

Raise $2.71828182$ to the power of $4.92842712$ .

$f'' (- 4.92842712) = \frac{24.28939392}{138.16202971} + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.6

Divide $24.28939392$ by $138.16202971$ .

$f'' (- 4.92842712) = 0.17580368 + 4 (- 4.92842712) \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.7

Multiply $4$ by $- 4.92842712$ .

$f'' (- 4.92842712) = 0.17580368 - 19.71370849 \cdot e^{- 4.92842712} - 4 e^{- 4.92842712}$

Step 6.2.1.8

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

$f'' (- 4.92842712) = 0.17580368 - 19.71370849 \cdot \frac{1}{e^{4.92842712}} - 4 e^{- 4.92842712}$

Step 6.2.1.9

Combine $- 19.71370849$ and $\frac{1}{e^{4.92842712}}$ .

$f'' (- 4.92842712) = 0.17580368 + \frac{- 19.71370849}{e^{4.92842712}} - 4 e^{- 4.92842712}$

Step 6.2.1.10

Move the negative in front of the fraction.

$f'' (- 4.92842712) = 0.17580368 - \frac{19.71370849}{e^{4.92842712}} - 4 e^{- 4.92842712}$

Step 6.2.1.11

Replace $e$ with an approximation.

$f'' (- 4.92842712) = 0.17580368 - \frac{19.71370849}{{2.71828182}^{4.92842712}} - 4 e^{- 4.92842712}$

Step 6.2.1.12

Raise $2.71828182$ to the power of $4.92842712$ .

$f'' (- 4.92842712) = 0.17580368 - \frac{19.71370849}{138.16202971} - 4 e^{- 4.92842712}$

Step 6.2.1.13

Divide $19.71370849$ by $138.16202971$ .

$f'' (- 4.92842712) = 0.17580368 - 1 \cdot 0.14268542 - 4 e^{- 4.92842712}$

Step 6.2.1.14

Multiply $- 1$ by $0.14268542$ .

$f'' (- 4.92842712) = 0.17580368 - 0.14268542 - 4 e^{- 4.92842712}$

Step 6.2.1.15

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

$f'' (- 4.92842712) = 0.17580368 - 0.14268542 - 4 \frac{1}{e^{4.92842712}}$

Step 6.2.1.16

Combine $- 4$ and $\frac{1}{e^{4.92842712}}$ .

$f'' (- 4.92842712) = 0.17580368 - 0.14268542 + \frac{- 4}{e^{4.92842712}}$

Step 6.2.1.17

Move the negative in front of the fraction.

$f'' (- 4.92842712) = 0.17580368 - 0.14268542 - \frac{4}{e^{4.92842712}}$

Step 6.2.2

Simplify by adding terms.

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Step 6.2.2.1

Subtract $0.14268542$ from $0.17580368$ .

$f'' (- 4.92842712) = 0.03311825 - \frac{4}{e^{4.92842712}}$

Step 6.2.2.2

Subtract $\frac{4}{e^{4.92842712}}$ from $0.03311825$ .

$f'' (- 4.92842712) = 0.00416674$

Step 6.2.3

The final answer is $0.00416674$ .

$0.00416674$

Step 6.3

At $- 4.92842712$ , the second derivative is $0.00416674$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2 - 2 \sqrt{2})$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Replace $e$ with an approximation.

$f'' (- 2) = \frac{4}{{2.71828182}^{2}} + 4 (- 2) \cdot e^{- 2} - 4 e^{- 2}$

Step 7.2.1.5

Raise $2.71828182$ to the power of $2$ .

$f'' (- 2) = \frac{4}{7.38905609} + 4 (- 2) \cdot e^{- 2} - 4 e^{- 2}$

Step 7.2.1.6

Divide $4$ by $7.38905609$ .

$f'' (- 2) = 0.54134113 + 4 (- 2) \cdot e^{- 2} - 4 e^{- 2}$

Step 7.2.1.7

Multiply $4$ by $- 2$ .

$f'' (- 2) = 0.54134113 - 8 \cdot e^{- 2} - 4 e^{- 2}$

Step 7.2.1.8

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

$f'' (- 2) = 0.54134113 - 8 \cdot \frac{1}{e^{2}} - 4 e^{- 2}$

Step 7.2.1.9

Combine $- 8$ and $\frac{1}{e^{2}}$ .

$f'' (- 2) = 0.54134113 + \frac{- 8}{e^{2}} - 4 e^{- 2}$

Step 7.2.1.10

Move the negative in front of the fraction.

$f'' (- 2) = 0.54134113 - \frac{8}{e^{2}} - 4 e^{- 2}$

Step 7.2.1.11

Replace $e$ with an approximation.

$f'' (- 2) = 0.54134113 - \frac{8}{{2.71828182}^{2}} - 4 e^{- 2}$

Step 7.2.1.12

Raise $2.71828182$ to the power of $2$ .

$f'' (- 2) = 0.54134113 - \frac{8}{7.38905609} - 4 e^{- 2}$

Step 7.2.1.13

Divide $8$ by $7.38905609$ .

$f'' (- 2) = 0.54134113 - 1 \cdot 1.08268226 - 4 e^{- 2}$

Step 7.2.1.14

Multiply $- 1$ by $1.08268226$ .

$f'' (- 2) = 0.54134113 - 1.08268226 - 4 e^{- 2}$

Step 7.2.1.15

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

$f'' (- 2) = 0.54134113 - 1.08268226 - 4 \frac{1}{e^{2}}$

Step 7.2.1.16

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

$f'' (- 2) = 0.54134113 - 1.08268226 + \frac{- 4}{e^{2}}$

Step 7.2.1.17

Move the negative in front of the fraction.

$f'' (- 2) = 0.54134113 - 1.08268226 - \frac{4}{e^{2}}$

Step 7.2.2

Simplify by adding terms.

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Step 7.2.2.1

Subtract $1.08268226$ from $0.54134113$ .

$f'' (- 2) = - 0.54134113 - \frac{4}{e^{2}}$

Step 7.2.2.2

Subtract $\frac{4}{e^{2}}$ from $- 0.54134113$ .

$f'' (- 2) = - 1.08268226$

Step 7.2.3

The final answer is $- 1.08268226$ .

$- 1.08268226$

Step 7.3

At $- 2$ , the second derivative is $- 1.08268226$ . Since this is negative, the second derivative is decreasing on the interval $(- 2 - 2 \sqrt{2}, - 2 + 2 \sqrt{2})$

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

Step 8

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

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Step 8.1

Replace the variable $x$ with $0.92842712$ in the expression.

$f'' (0.92842712) = {(0.92842712)}^{2} e^{0.92842712} + 4 (0.92842712) \cdot e^{0.92842712} - 4 e^{0.92842712}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $0.92842712$ to the power of $2$ .

$f'' (0.92842712) = 0.86197692 e^{0.92842712} + 4 (0.92842712) \cdot e^{0.92842712} - 4 e^{0.92842712}$

Step 8.2.1.2

Multiply $0.86197692$ by $e^{0.92842712}$ .

$f'' (0.92842712) = 2.18125488 + 4 (0.92842712) \cdot e^{0.92842712} - 4 e^{0.92842712}$

Step 8.2.1.3

Multiply $4$ by $0.92842712$ .

$f'' (0.92842712) = 2.18125488 + 3.71370849 \cdot e^{0.92842712} - 4 e^{0.92842712}$

Step 8.2.1.4

Multiply $3.71370849$ by $e^{0.92842712}$ .

$f'' (0.92842712) = 2.18125488 + 9.39763533 - 4 e^{0.92842712}$

Step 8.2.2

Simplify by adding terms.

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Step 8.2.2.1

Add $2.18125488$ and $9.39763533$ .

$f'' (0.92842712) = 11.57889022 - 4 e^{0.92842712}$

Step 8.2.2.2

Subtract $4 e^{0.92842712}$ from $11.57889022$ .

$f'' (0.92842712) = 1.45678684$

Step 8.2.3

The final answer is $1.45678684$ .

$1.45678684$

Step 8.3

At $0.92842712$ , the second derivative is $1.45678684$ . Since this is positive, the second derivative is increasing on the interval $(- 2 + 2 \sqrt{2}, \infty)$ .

Increasing on $(- 2 + 2 \sqrt{2}, \infty)$ 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 $(- 2 - 2 \sqrt{2}, 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}}), (- 2 + 2 \sqrt{2}, 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}})$ .

$(- 2 - 2 \sqrt{2}, 6 e^{- 2 - 2 \sqrt{2}} + 8 \sqrt{2} e^{- 2 - 2 \sqrt{2}}), (- 2 + 2 \sqrt{2}, 6 e^{- 2 + 2 \sqrt{2}} - 8 \sqrt{2} e^{- 2 + 2 \sqrt{2}})$

Step 10