Calculus Examples

$y = x^{2} e^{x}$

Step 1

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

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

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Simplify.

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Step 2.1.4.1

Reorder terms.

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

Step 2.1.4.2

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

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

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

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

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 $a$ = $e$ .

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

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 $n = 2$ .

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

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

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

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

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

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 $a$ = $e$ .

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

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 $n = 1$ .

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

Step 2.2.3.5

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

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

Step 2.2.4

Simplify.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

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

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Step 2.2.4.2.1

Move $e^{x}$ .

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

Step 2.2.4.2.2

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

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

Step 2.2.4.3

Reorder terms.

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

Step 2.2.4.4

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

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 $0$ .

$e^{x} = 0$

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

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

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Step 3.5.1

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

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

Step 3.5.2

Solve $x^{2} + 4 x + 2 = 0$ 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 $c = 2$ into the quadratic formula and solve for $x$ .

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

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 $2$ .

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

Step 3.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 3.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.5.2.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Step 3.5.2.3.1.3

Subtract $8$ from $16$ .

$x = \frac{- 4 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.5.2.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.5.2.3.1.4.1

Factor $4$ out of $8$ .

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

Step 3.5.2.3.1.4.2

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

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

Step 3.5.2.3.1.5

Pull terms out from under the radical.

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

Step 3.5.2.3.2

Multiply $2$ by $1$ .

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

Step 3.5.2.3.3

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

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

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 $2$ .

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

Step 3.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 3.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.5.2.4.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Step 3.5.2.4.1.3

Subtract $8$ from $16$ .

$x = \frac{- 4 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.5.2.4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.5.2.4.1.4.1

Factor $4$ out of $8$ .

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

Step 3.5.2.4.1.4.2

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

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

Step 3.5.2.4.1.5

Pull terms out from under the radical.

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

Step 3.5.2.4.2

Multiply $2$ by $1$ .

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

Step 3.5.2.4.3

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

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

Step 3.5.2.4.4

Change the $\pm$ to $+$ .

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

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 $2$ .

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

Step 3.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 3.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.5.2.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Step 3.5.2.5.1.3

Subtract $8$ from $16$ .

$x = \frac{- 4 \pm \sqrt{8}}{2 \cdot 1}$

Step 3.5.2.5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.5.2.5.1.4.1

Factor $4$ out of $8$ .

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

Step 3.5.2.5.1.4.2

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

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

Step 3.5.2.5.1.5

Pull terms out from under the radical.

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

Step 3.5.2.5.2

Multiply $2$ by $1$ .

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

Step 3.5.2.5.3

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

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

Step 3.5.2.5.4

Change the $\pm$ to $-$ .

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

Step 3.5.2.6

The final answer is the combination of both solutions.

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

Step 3.6

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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Step 4.1.2.2.1

Apply the distributive property.

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

Step 4.1.2.2.2

Apply the distributive property.

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

Step 4.1.2.2.3

Apply the distributive property.

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

Step 4.1.2.3

Simplify and combine like terms.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Multiply $- 2$ by $- 2$ .

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

Step 4.1.2.3.1.2

Move $- 2$ to the left of $\sqrt{2}$ .

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

Step 4.1.2.3.1.3

Combine using the product rule for radicals.

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

Step 4.1.2.3.1.4

Multiply $2$ by $2$ .

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

Step 4.1.2.3.1.5

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

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

Step 4.1.2.3.1.6

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

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

Step 4.1.2.3.2

Add $4$ and $2$ .

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

Step 4.1.2.3.3

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

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

Step 4.1.2.4

Apply the distributive property.

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

Step 4.1.2.5

The final answer is $6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}}$ .

$6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}}$

Step 4.2

The point found by substituting $- 2 + \sqrt{2}$ in $f (x) = x^{2} e^{x}$ is $(- 2 + \sqrt{2}, 6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}})$ . This point can be an inflection point.

$(- 2 + \sqrt{2}, 6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}})$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

Apply the distributive property.

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

Step 4.3.2.2.2

Apply the distributive property.

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

Step 4.3.2.2.3

Apply the distributive property.

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

Step 4.3.2.3

Simplify and combine like terms.

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Step 4.3.2.3.1

Simplify each term.

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Step 4.3.2.3.1.1

Multiply $- 2$ by $- 2$ .

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

Step 4.3.2.3.1.2

Multiply $- 1$ by $- 2$ .

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

Step 4.3.2.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 4.3.2.3.1.4

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

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Step 4.3.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.3.1.4.2

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

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

Step 4.3.2.3.1.4.3

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

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

Step 4.3.2.3.1.4.4

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

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

Step 4.3.2.3.1.4.5

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

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

Step 4.3.2.3.1.4.6

Add $1$ and $1$ .

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

Step 4.3.2.3.1.5

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

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Step 4.3.2.3.1.5.1

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

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

Step 4.3.2.3.1.5.2

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

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

Step 4.3.2.3.1.5.3

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

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

Step 4.3.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.3.1.5.4.1

Cancel the common factor.

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

Step 4.3.2.3.1.5.4.2

Rewrite the expression.

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

Step 4.3.2.3.1.5.5

Evaluate the exponent.

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

Step 4.3.2.3.2

Add $4$ and $2$ .

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

Step 4.3.2.3.3

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

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

Step 4.3.2.4

Apply the distributive property.

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

Step 4.3.2.5

The final answer is $6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}}$ .

$6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}}$

Step 4.4

The point found by substituting $- 2 - \sqrt{2}$ in $f (x) = x^{2} e^{x}$ is $(- 2 - \sqrt{2}, 6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}})$ . This point can be an inflection point.

$(- 2 - \sqrt{2}, 6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}})$

Step 4.5

Determine the points that could be inflection points.

$(- 2 + \sqrt{2}, 6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}}), (- 2 - \sqrt{2}, 6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}})$

Step 5

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

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 3.51421356$ in the expression.

$f'' (- 3.51421356) = {(- 3.51421356)}^{2} e^{- 3.51421356} + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

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 $- 3.51421356$ to the power of $2$ .

$f'' (- 3.51421356) = 12.34969696 e^{- 3.51421356} + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.2

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

$f'' (- 3.51421356) = 12.34969696 (\frac{1}{e^{3.51421356}}) + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.3

Combine $12.34969696$ and $\frac{1}{e^{3.51421356}}$ .

$f'' (- 3.51421356) = \frac{12.34969696}{e^{3.51421356}} + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.4

Replace $e$ with an approximation.

$f'' (- 3.51421356) = \frac{12.34969696}{{2.71828182}^{3.51421356}} + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.5

Raise $2.71828182$ to the power of $3.51421356$ .

$f'' (- 3.51421356) = \frac{12.34969696}{33.58950148} + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.6

Divide $12.34969696$ by $33.58950148$ .

$f'' (- 3.51421356) = 0.36766538 + 4 (- 3.51421356) \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.7

Multiply $4$ by $- 3.51421356$ .

$f'' (- 3.51421356) = 0.36766538 - 14.05685424 \cdot e^{- 3.51421356} + 2 e^{- 3.51421356}$

Step 6.2.1.8

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

$f'' (- 3.51421356) = 0.36766538 - 14.05685424 \cdot \frac{1}{e^{3.51421356}} + 2 e^{- 3.51421356}$

Step 6.2.1.9

Combine $- 14.05685424$ and $\frac{1}{e^{3.51421356}}$ .

$f'' (- 3.51421356) = 0.36766538 + \frac{- 14.05685424}{e^{3.51421356}} + 2 e^{- 3.51421356}$

Step 6.2.1.10

Move the negative in front of the fraction.

$f'' (- 3.51421356) = 0.36766538 - \frac{14.05685424}{e^{3.51421356}} + 2 e^{- 3.51421356}$

Step 6.2.1.11

Replace $e$ with an approximation.

$f'' (- 3.51421356) = 0.36766538 - \frac{14.05685424}{{2.71828182}^{3.51421356}} + 2 e^{- 3.51421356}$

Step 6.2.1.12

Raise $2.71828182$ to the power of $3.51421356$ .

$f'' (- 3.51421356) = 0.36766538 - \frac{14.05685424}{33.58950148} + 2 e^{- 3.51421356}$

Step 6.2.1.13

Divide $14.05685424$ by $33.58950148$ .

$f'' (- 3.51421356) = 0.36766538 - 1 \cdot 0.41848951 + 2 e^{- 3.51421356}$

Step 6.2.1.14

Multiply $- 1$ by $0.41848951$ .

$f'' (- 3.51421356) = 0.36766538 - 0.41848951 + 2 e^{- 3.51421356}$

Step 6.2.1.15

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

$f'' (- 3.51421356) = 0.36766538 - 0.41848951 + 2 (\frac{1}{e^{3.51421356}})$

Step 6.2.1.16

Combine $2$ and $\frac{1}{e^{3.51421356}}$ .

$f'' (- 3.51421356) = 0.36766538 - 0.41848951 + \frac{2}{e^{3.51421356}}$

Step 6.2.2

Simplify by adding terms.

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Step 6.2.2.1

Subtract $0.41848951$ from $0.36766538$ .

$f'' (- 3.51421356) = - 0.05082413 + \frac{2}{e^{3.51421356}}$

Step 6.2.2.2

Add $- 0.05082413$ and $\frac{2}{e^{3.51421356}}$ .

$f'' (- 3.51421356) = 0.00871828$

Step 6.2.3

The final answer is $0.00871828$ .

$0.00871828$

Step 6.3

At $- 3.51421356$ , the second derivative is $0.00871828$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2 - \sqrt{2})$ .

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

Step 7

Substitute a value from the interval $(- 2 - \sqrt{2}, - 2 + \sqrt{2})$ 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.

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

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 $2$ .

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Replace $e$ with an approximation.

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

Step 7.2.1.5

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

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

Step 7.2.1.6

Divide $4$ by $7.38905609$ .

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

Step 7.2.1.7

Multiply $4$ by $- 2$ .

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

Step 7.2.1.8

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

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

Step 7.2.1.9

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

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

Step 7.2.1.10

Move the negative in front of the fraction.

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

Step 7.2.1.11

Replace $e$ with an approximation.

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

Step 7.2.1.12

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

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

Step 7.2.1.13

Divide $8$ by $7.38905609$ .

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

Step 7.2.1.14

Multiply $- 1$ by $1.08268226$ .

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

Step 7.2.1.15

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

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

Step 7.2.1.16

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

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

Step 7.2.2

Simplify by adding terms.

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Step 7.2.2.1

Subtract $1.08268226$ from $0.54134113$ .

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

Step 7.2.2.2

Add $- 0.54134113$ and $\frac{2}{e^{2}}$ .

$f'' (- 2) = - 0.27067056$

Step 7.2.3

The final answer is $- 0.27067056$ .

$- 0.27067056$

Step 7.3

At $- 2$ , the second derivative is $- 0.27067056$ . Since this is negative, the second derivative is decreasing on the interval $(- 2 - \sqrt{2}, - 2 + \sqrt{2})$

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

Step 8

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

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Step 8.1

Replace the variable $x$ with $- 0.48578643$ in the expression.

$f'' (- 0.48578643) = {(- 0.48578643)}^{2} e^{- 0.48578643} + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

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 $- 0.48578643$ to the power of $2$ .

$f'' (- 0.48578643) = 0.23598846 e^{- 0.48578643} + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.2

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

$f'' (- 0.48578643) = 0.23598846 (\frac{1}{e^{0.48578643}}) + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.3

Combine $0.23598846$ and $\frac{1}{e^{0.48578643}}$ .

$f'' (- 0.48578643) = \frac{0.23598846}{e^{0.48578643}} + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.4

Replace $e$ with an approximation.

$f'' (- 0.48578643) = \frac{0.23598846}{{2.71828182}^{0.48578643}} + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.5

Raise $2.71828182$ to the power of $0.48578643$ .

$f'' (- 0.48578643) = \frac{0.23598846}{1.62545282} + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.6

Divide $0.23598846$ by $1.62545282$ .

$f'' (- 0.48578643) = 0.14518321 + 4 (- 0.48578643) \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.7

Multiply $4$ by $- 0.48578643$ .

$f'' (- 0.48578643) = 0.14518321 - 1.94314575 \cdot e^{- 0.48578643} + 2 e^{- 0.48578643}$

Step 8.2.1.8

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

$f'' (- 0.48578643) = 0.14518321 - 1.94314575 \cdot \frac{1}{e^{0.48578643}} + 2 e^{- 0.48578643}$

Step 8.2.1.9

Combine $- 1.94314575$ and $\frac{1}{e^{0.48578643}}$ .

$f'' (- 0.48578643) = 0.14518321 + \frac{- 1.94314575}{e^{0.48578643}} + 2 e^{- 0.48578643}$

Step 8.2.1.10

Move the negative in front of the fraction.

$f'' (- 0.48578643) = 0.14518321 - \frac{1.94314575}{e^{0.48578643}} + 2 e^{- 0.48578643}$

Step 8.2.1.11

Replace $e$ with an approximation.

$f'' (- 0.48578643) = 0.14518321 - \frac{1.94314575}{{2.71828182}^{0.48578643}} + 2 e^{- 0.48578643}$

Step 8.2.1.12

Raise $2.71828182$ to the power of $0.48578643$ .

$f'' (- 0.48578643) = 0.14518321 - \frac{1.94314575}{1.62545282} + 2 e^{- 0.48578643}$

Step 8.2.1.13

Divide $1.94314575$ by $1.62545282$ .

$f'' (- 0.48578643) = 0.14518321 - 1 \cdot 1.19544887 + 2 e^{- 0.48578643}$

Step 8.2.1.14

Multiply $- 1$ by $1.19544887$ .

$f'' (- 0.48578643) = 0.14518321 - 1.19544887 + 2 e^{- 0.48578643}$

Step 8.2.1.15

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

$f'' (- 0.48578643) = 0.14518321 - 1.19544887 + 2 (\frac{1}{e^{0.48578643}})$

Step 8.2.1.16

Combine $2$ and $\frac{1}{e^{0.48578643}}$ .

$f'' (- 0.48578643) = 0.14518321 - 1.19544887 + \frac{2}{e^{0.48578643}}$

Step 8.2.2

Simplify by adding terms.

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Step 8.2.2.1

Subtract $1.19544887$ from $0.14518321$ .

$f'' (- 0.48578643) = - 1.05026566 + \frac{2}{e^{0.48578643}}$

Step 8.2.2.2

Add $- 1.05026566$ and $\frac{2}{e^{0.48578643}}$ .

$f'' (- 0.48578643) = 0.18016069$

Step 8.2.3

The final answer is $0.18016069$ .

$0.18016069$

Step 8.3

At $- 0.48578643$ , the second derivative is $0.18016069$ . Since this is positive, the second derivative is increasing on the interval $(- 2 + \sqrt{2}, \infty)$ .

Increasing on $(- 2 + \sqrt{2}, \infty)$ 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 $(- 2 - \sqrt{2}, 6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}}), (- 2 + \sqrt{2}, 6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}})$ .

$(- 2 - \sqrt{2}, 6 e^{- 2 - \sqrt{2}} + 4 \sqrt{2} e^{- 2 - \sqrt{2}}), (- 2 + \sqrt{2}, 6 e^{- 2 + \sqrt{2}} - 4 \sqrt{2} e^{- 2 + \sqrt{2}})$

Step 10