Calculus Examples

$f (x) = x^{\frac{2}{3}} e^{x}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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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 $f (x) = x^{\frac{2}{3}}$ and $g (x) = e^{x}$ .

$x^{\frac{2}{3}} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 1.1.2

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

$x^{\frac{2}{3}} e^{x} + e^{x} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$x^{\frac{2}{3}} e^{x} + e^{x} (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 1.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.1.5

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $3$ .

$x^{\frac{2}{3}} e^{x} + e^{x} (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 1.1.7.2

Subtract $3$ from $2$ .

$x^{\frac{2}{3}} e^{x} + e^{x} (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 1.1.8

Move the negative in front of the fraction.

$x^{\frac{2}{3}} e^{x} + e^{x} (\frac{2}{3} x^{- \frac{1}{3}})$

Step 1.1.9

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

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

Step 1.1.10

Combine $e^{x}$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$x^{\frac{2}{3}} e^{x} + \frac{e^{x} (2 x^{- \frac{1}{3}})}{3}$

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{2}{3}} e^{x} + \frac{e^{x} \cdot 2}{3 x^{\frac{1}{3}}}$

Step 1.1.11.2

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

$x^{\frac{2}{3}} e^{x} + \frac{2 \cdot e^{x}}{3 x^{\frac{1}{3}}}$

Step 1.1.11.3

Reorder terms.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $e^{x} x^{\frac{2}{3}} + \frac{2 e^{x}}{3 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{d}{d x} [e^{x} x^{\frac{2}{3}}] + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$ .

$\frac{d}{d x} [e^{x} x^{\frac{2}{3}}] + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [e^{x} x^{\frac{2}{3}}]$ .

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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 $f (x) = e^{x}$ and $g (x) = x^{\frac{2}{3}}$ .

$e^{x} \frac{d}{d x} [x^{\frac{2}{3}}] + x^{\frac{2}{3}} \frac{d}{d x} [e^{x}] + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$e^{x} (\frac{2}{3} x^{\frac{2}{3} - 1}) + x^{\frac{2}{3}} \frac{d}{d x} [e^{x}] + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.3

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

$e^{x} (\frac{2}{3} x^{\frac{2}{3} - 1}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e^{x} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.5

Combine $- 1$ and $\frac{3}{3}$ .

$e^{x} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.6

Combine the numerators over the common denominator.

$e^{x} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.7

Simplify the numerator.

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Step 1.2.2.7.1

Multiply $- 1$ by $3$ .

$e^{x} (\frac{2}{3} x^{\frac{2 - 3}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.7.2

Subtract $3$ from $2$ .

$e^{x} (\frac{2}{3} x^{\frac{- 1}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.8

Move the negative in front of the fraction.

$e^{x} (\frac{2}{3} x^{- \frac{1}{3}}) + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.9

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$e^{x} \frac{2 x^{- \frac{1}{3}}}{3} + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.10

Combine $e^{x}$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{e^{x} (2 x^{- \frac{1}{3}})}{3} + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.11

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{e^{x} \cdot 2}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.2.12

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

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [\frac{2 e^{x}}{3 x^{\frac{1}{3}}}]$ .

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Step 1.2.3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 e^{x}}{3 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\frac{e^{x}}{x^{\frac{1}{3}}}]$ .

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{2}{3} \frac{d}{d x} [\frac{e^{x}}{x^{\frac{1}{3}}}]$

Step 1.2.3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = x^{\frac{1}{3}}$ .

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{2}{3} \cdot \frac{x^{\frac{1}{3}} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{\frac{1}{3}}]}{{(x^{\frac{1}{3}})}^{2}}$

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 $a$ = $e$ .

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{2}{3} \cdot \frac{x^{\frac{1}{3}} e^{x} - e^{x} \frac{d}{d x} [x^{\frac{1}{3}}]}{{(x^{\frac{1}{3}})}^{2}}$

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 $n = \frac{1}{3}$ .

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

Step 1.2.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.2.3.6

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 1.2.3.7

Combine the numerators over the common denominator.

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

Step 1.2.3.8

Simplify the numerator.

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Step 1.2.3.8.1

Multiply $- 1$ by $3$ .

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

Step 1.2.3.8.2

Subtract $3$ from $1$ .

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

Step 1.2.3.9

Move the negative in front of the fraction.

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

Step 1.2.3.10

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

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

Step 1.2.3.11

Combine $\frac{x^{- \frac{2}{3}}}{3}$ and $e^{x}$ .

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

Step 1.2.3.12

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2.3.13

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 1.2.3.13.1

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

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

Step 1.2.3.13.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 1.2.3.14

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

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{2 (x^{\frac{1}{3}} e^{x} - \frac{e^{x}}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Apply the distributive property.

$\frac{2 e^{x}}{3 x^{\frac{1}{3}}} + x^{\frac{2}{3}} e^{x} + \frac{2 (x^{\frac{1}{3}} e^{x}) + 2 (- \frac{e^{x}}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2

Combine terms.

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Step 1.2.4.2.1

Multiply $- 1$ by $2$ .

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

Step 1.2.4.2.2

Combine $- 2$ and $\frac{e^{x}}{3 x^{\frac{2}{3}}}$ .

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

Step 1.2.4.2.3

Move the negative in front of the fraction.

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

Step 1.2.4.2.4

To write $\frac{2 e^{x}}{3 x^{\frac{1}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

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

Step 1.2.4.2.5

Write each expression with a common denominator of $3 x^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.2.5.1

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

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

Step 1.2.4.2.5.2

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 1.2.4.2.5.2.1

Move $x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} e^{x} + \frac{2 e^{x} x^{\frac{1}{3}}}{3 (x^{\frac{1}{3}} x^{\frac{1}{3}})} + \frac{2 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.5.2.2

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

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

Step 1.2.4.2.5.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4.2.5.2.4

Add $1$ and $1$ .

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

Step 1.2.4.2.6

Combine the numerators over the common denominator.

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

Step 1.2.4.2.7

Add $2 e^{x} x^{\frac{1}{3}}$ and $2 x^{\frac{1}{3}} e^{x}$ .

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Step 1.2.4.2.7.1

Move $e^{x}$ .

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

Step 1.2.4.2.7.2

Add $2 x^{\frac{1}{3}} e^{x}$ and $2 x^{\frac{1}{3}} e^{x}$ .

$x^{\frac{2}{3}} e^{x} + \frac{4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.8

To write $x^{\frac{2}{3}} e^{x}$ as a fraction with a common denominator, multiply by $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$x^{\frac{2}{3}} e^{x} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}} + \frac{4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.9

Combine $x^{\frac{2}{3}} e^{x}$ and $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$\frac{x^{\frac{2}{3}} e^{x} (3 x^{\frac{2}{3}})}{3 x^{\frac{2}{3}}} + \frac{4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.10

Combine the numerators over the common denominator.

$\frac{x^{\frac{2}{3}} e^{x} (3 x^{\frac{2}{3}}) + 4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.11

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

$\frac{x^{\frac{2}{3} + \frac{2}{3}} e^{x} \cdot 3 + 4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.12

Combine the numerators over the common denominator.

$\frac{x^{\frac{2 + 2}{3}} e^{x} \cdot 3 + 4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.13

Add $2$ and $2$ .

$\frac{x^{\frac{4}{3}} e^{x} \cdot 3 + 4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.2.14

Move $3$ to the left of $x^{\frac{4}{3}} e^{x}$ .

$\frac{3 x^{\frac{4}{3}} e^{x} + 4 x^{\frac{1}{3}} e^{x} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.3

Reorder terms.

$\frac{3 e^{x} x^{\frac{4}{3}} + 4 e^{x} x^{\frac{1}{3}} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4

Simplify the numerator.

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Step 1.2.4.4.1

Factor $e^{x}$ out of $3 e^{x} x^{\frac{4}{3}} + 4 e^{x} x^{\frac{1}{3}} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}$ .

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Step 1.2.4.4.1.1

Factor $e^{x}$ out of $3 e^{x} x^{\frac{4}{3}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}}) + 4 e^{x} x^{\frac{1}{3}} - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.1.2

Factor $e^{x}$ out of $4 e^{x} x^{\frac{1}{3}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}}) + e^{x} (4 x^{\frac{1}{3}}) - \frac{2 e^{x}}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.1.3

Factor $e^{x}$ out of $- \frac{2 e^{x}}{3 x^{\frac{2}{3}}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}}) + e^{x} (4 x^{\frac{1}{3}}) + e^{x} (- \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.1.4

Factor $e^{x}$ out of $e^{x} (3 x^{\frac{4}{3}}) + e^{x} (4 x^{\frac{1}{3}})$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + 4 x^{\frac{1}{3}}) + e^{x} (- \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.1.5

Factor $e^{x}$ out of $e^{x} (3 x^{\frac{4}{3}} + 4 x^{\frac{1}{3}}) + e^{x} (- \frac{2}{3 x^{\frac{2}{3}}})$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + 4 x^{\frac{1}{3}} - \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.2

Reorder terms.

$\frac{e^{x} (4 x^{\frac{1}{3}} + 3 x^{\frac{4}{3}} - \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.3

To write $4 x^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + 4 x^{\frac{1}{3}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}} - \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.4

Combine $4 x^{\frac{1}{3}}$ and $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{4 x^{\frac{1}{3}} (3 x^{\frac{2}{3}})}{3 x^{\frac{2}{3}}} - \frac{2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.5

Combine the numerators over the common denominator.

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{4 x^{\frac{1}{3}} (3 x^{\frac{2}{3}}) - 2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6

Simplify the numerator.

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Step 1.2.4.4.6.1

Factor $2$ out of $4 x^{\frac{1}{3}} (3 x^{\frac{2}{3}}) - 2$ .

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Step 1.2.4.4.6.1.1

Factor $2$ out of $4 x^{\frac{1}{3}} (3 x^{\frac{2}{3}})$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 x^{\frac{1}{3}} (3 x^{\frac{2}{3}})) - 2}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.1.2

Factor $2$ out of $- 2$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 x^{\frac{1}{3}} (3 x^{\frac{2}{3}})) + 2 (- 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.1.3

Factor $2$ out of $2 (2 x^{\frac{1}{3}} (3 x^{\frac{2}{3}})) + 2 (- 1)$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 x^{\frac{1}{3}} (3 x^{\frac{2}{3}}) - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.2

Rewrite using the commutative property of multiplication.

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x^{\frac{1}{3}} x^{\frac{2}{3}} - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.3

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.2.4.4.6.3.1

Move $x^{\frac{2}{3}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 (x^{\frac{2}{3}} x^{\frac{1}{3}}) - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.3.2

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

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x^{\frac{2}{3} + \frac{1}{3}} - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.3.3

Combine the numerators over the common denominator.

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x^{\frac{2 + 1}{3}} - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.3.4

Add $2$ and $1$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x^{\frac{3}{3}} - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.3.5

Divide $3$ by $3$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x^{1} - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.4

Simplify $2 \cdot 3 x^{1}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (2 \cdot 3 x - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.6.5

Multiply $2$ by $3$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} + \frac{2 (6 x - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.7

To write $3 x^{\frac{4}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$\frac{e^{x} (3 x^{\frac{4}{3}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}} + \frac{2 (6 x - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.8

Combine $3 x^{\frac{4}{3}}$ and $\frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ .

$\frac{e^{x} (\frac{3 x^{\frac{4}{3}} (3 x^{\frac{2}{3}})}{3 x^{\frac{2}{3}}} + \frac{2 (6 x - 1)}{3 x^{\frac{2}{3}}})}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.9

Combine the numerators over the common denominator.

$\frac{e^{x} \frac{3 x^{\frac{4}{3}} (3 x^{\frac{2}{3}}) + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10

Simplify the numerator.

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Step 1.2.4.4.10.1

Rewrite using the commutative property of multiplication.

$\frac{e^{x} \frac{3 \cdot 3 x^{\frac{4}{3}} x^{\frac{2}{3}} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.2

Multiply $x^{\frac{4}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.2.4.4.10.2.1

Move $x^{\frac{2}{3}}$ .

$\frac{e^{x} \frac{3 \cdot 3 (x^{\frac{2}{3}} x^{\frac{4}{3}}) + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.2.2

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

$\frac{e^{x} \frac{3 \cdot 3 x^{\frac{2}{3} + \frac{4}{3}} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.2.3

Combine the numerators over the common denominator.

$\frac{e^{x} \frac{3 \cdot 3 x^{\frac{2 + 4}{3}} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.2.4

Add $2$ and $4$ .

$\frac{e^{x} \frac{3 \cdot 3 x^{\frac{6}{3}} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.2.5

Divide $6$ by $3$ .

$\frac{e^{x} \frac{3 \cdot 3 x^{2} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.3

Multiply $3$ by $3$ .

$\frac{e^{x} \frac{9 x^{2} + 2 (6 x - 1)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.4

Apply the distributive property.

$\frac{e^{x} \frac{9 x^{2} + 2 (6 x) + 2 \cdot - 1}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.5

Multiply $6$ by $2$ .

$\frac{e^{x} \frac{9 x^{2} + 12 x + 2 \cdot - 1}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.4.10.6

Multiply $2$ by $- 1$ .

$\frac{e^{x} \frac{9 x^{2} + 12 x - 2}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.5

Combine $e^{x}$ and $\frac{9 x^{2} + 12 x - 2}{3 x^{\frac{2}{3}}}$ .

$\frac{\frac{e^{x} (9 x^{2} + 12 x - 2)}{3 x^{\frac{2}{3}}}}{3 x^{\frac{2}{3}}}$

Step 1.2.4.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{e^{x} (9 x^{2} + 12 x - 2)}{3 x^{\frac{2}{3}}} \cdot \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.2.4.7

Combine.

$\frac{e^{x} (9 x^{2} + 12 x - 2) \cdot 1}{3 x^{\frac{2}{3}} (3 x^{\frac{2}{3}})}$

Step 1.2.4.8

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.2.4.8.1

Move $x^{\frac{2}{3}}$ .

$\frac{e^{x} (9 x^{2} + 12 x - 2) \cdot 1}{3 (x^{\frac{2}{3}} x^{\frac{2}{3}}) \cdot 3}$

Step 1.2.4.8.2

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

$\frac{e^{x} (9 x^{2} + 12 x - 2) \cdot 1}{3 x^{\frac{2}{3} + \frac{2}{3}} \cdot 3}$

Step 1.2.4.8.3

Combine the numerators over the common denominator.

$\frac{e^{x} (9 x^{2} + 12 x - 2) \cdot 1}{3 x^{\frac{2 + 2}{3}} \cdot 3}$

Step 1.2.4.8.4

Add $2$ and $2$ .

$\frac{e^{x} (9 x^{2} + 12 x - 2) \cdot 1}{3 x^{\frac{4}{3}} \cdot 3}$

Step 1.2.4.9

Multiply $e^{x} (9 x^{2} + 12 x - 2)$ by $1$ .

$\frac{e^{x} (9 x^{2} + 12 x - 2)}{3 x^{\frac{4}{3}} \cdot 3}$

Step 1.2.4.10

Multiply $3$ by $3$ .

$f'' (x) = \frac{e^{x} (9 x^{2} + 12 x - 2)}{9 x^{\frac{4}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{e^{x} (9 x^{2} + 12 x - 2)}{9 x^{\frac{4}{3}}}$ .

$\frac{e^{x} (9 x^{2} + 12 x - 2)}{9 x^{\frac{4}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{e^{x} (9 x^{2} + 12 x - 2)}{9 x^{\frac{4}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{e^{x} (9 x^{2} + 12 x - 2)}{9 x^{\frac{4}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$e^{x} (9 x^{2} + 12 x - 2) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$e^{x} = 0$

$9 x^{2} + 12 x - 2 = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$e^{x} = 0$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

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

Undefined

Step 2.3.2.2.3

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

No solution

Step 2.3.3

Set $9 x^{2} + 12 x - 2$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $9 x^{2} + 12 x - 2$ equal to $0$ .

$9 x^{2} + 12 x - 2 = 0$

Step 2.3.3.2

Solve $9 x^{2} + 12 x - 2 = 0$ for $x$ .

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Step 2.3.3.2.1

Use the quadratic formula to find the solutions.

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

Step 2.3.3.2.2

Substitute the values $a = 9$ , $b = 12$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (9 \cdot - 2)}}{2 \cdot 9}$

Step 2.3.3.2.3

Simplify.

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Step 2.3.3.2.3.1

Simplify the numerator.

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Step 2.3.3.2.3.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.3.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 2.3.3.2.3.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.3.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 72}}{2 \cdot 9}$

Step 2.3.3.2.3.1.3

Add $144$ and $72$ .

$x = \frac{- 12 \pm \sqrt{216}}{2 \cdot 9}$

Step 2.3.3.2.3.1.4

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 2.3.3.2.3.1.4.1

Factor $36$ out of $216$ .

$x = \frac{- 12 \pm \sqrt{36 (6)}}{2 \cdot 9}$

Step 2.3.3.2.3.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} \cdot 6}}{2 \cdot 9}$

Step 2.3.3.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{6}}{2 \cdot 9}$

Step 2.3.3.2.3.2

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 6 \sqrt{6}}{18}$

Step 2.3.3.2.3.3

Simplify $\frac{- 12 \pm 6 \sqrt{6}}{18}$ .

$x = \frac{- 2 \pm \sqrt{6}}{3}$

Step 2.3.3.2.4

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

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Step 2.3.3.2.4.1

Simplify the numerator.

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Step 2.3.3.2.4.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.4.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 2.3.3.2.4.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.4.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 72}}{2 \cdot 9}$

Step 2.3.3.2.4.1.3

Add $144$ and $72$ .

$x = \frac{- 12 \pm \sqrt{216}}{2 \cdot 9}$

Step 2.3.3.2.4.1.4

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 2.3.3.2.4.1.4.1

Factor $36$ out of $216$ .

$x = \frac{- 12 \pm \sqrt{36 (6)}}{2 \cdot 9}$

Step 2.3.3.2.4.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} \cdot 6}}{2 \cdot 9}$

Step 2.3.3.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{6}}{2 \cdot 9}$

Step 2.3.3.2.4.2

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 6 \sqrt{6}}{18}$

Step 2.3.3.2.4.3

Simplify $\frac{- 12 \pm 6 \sqrt{6}}{18}$ .

$x = \frac{- 2 \pm \sqrt{6}}{3}$

Step 2.3.3.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + \sqrt{6}}{3}$

Step 2.3.3.2.4.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + \sqrt{6}}{3}$

Step 2.3.3.2.4.6

Factor $- 1$ out of $\sqrt{6}$ .

$x = \frac{- 1 \cdot 2 - 1 (- \sqrt{6})}{3}$

Step 2.3.3.2.4.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{6})$ .

$x = \frac{- 1 (2 - \sqrt{6})}{3}$

Step 2.3.3.2.4.8

Move the negative in front of the fraction.

$x = - \frac{2 - \sqrt{6}}{3}$

Step 2.3.3.2.5

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

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Step 2.3.3.2.5.1

Simplify the numerator.

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Step 2.3.3.2.5.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.5.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 2.3.3.2.5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot - 2}}{2 \cdot 9}$

Step 2.3.3.2.5.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 72}}{2 \cdot 9}$

Step 2.3.3.2.5.1.3

Add $144$ and $72$ .

$x = \frac{- 12 \pm \sqrt{216}}{2 \cdot 9}$

Step 2.3.3.2.5.1.4

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 2.3.3.2.5.1.4.1

Factor $36$ out of $216$ .

$x = \frac{- 12 \pm \sqrt{36 (6)}}{2 \cdot 9}$

Step 2.3.3.2.5.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} \cdot 6}}{2 \cdot 9}$

Step 2.3.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{6}}{2 \cdot 9}$

Step 2.3.3.2.5.2

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 6 \sqrt{6}}{18}$

Step 2.3.3.2.5.3

Simplify $\frac{- 12 \pm 6 \sqrt{6}}{18}$ .

$x = \frac{- 2 \pm \sqrt{6}}{3}$

Step 2.3.3.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - \sqrt{6}}{3}$

Step 2.3.3.2.5.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - \sqrt{6}}{3}$

Step 2.3.3.2.5.6

Factor $- 1$ out of $- \sqrt{6}$ .

$x = \frac{- 1 \cdot 2 - (\sqrt{6})}{3}$

Step 2.3.3.2.5.7

Factor $- 1$ out of $- 1 (2) - (\sqrt{6})$ .

$x = \frac{- 1 (2 + \sqrt{6})}{3}$

Step 2.3.3.2.5.8

Move the negative in front of the fraction.

$x = - \frac{2 + \sqrt{6}}{3}$

Step 2.3.3.2.6

The final answer is the combination of both solutions.

$x = - \frac{2 - \sqrt{6}}{3}, - \frac{2 + \sqrt{6}}{3}$

Step 2.3.4

The final solution is all the values that make $e^{x} (9 x^{2} + 12 x - 2) = 0$ true.

$x = - \frac{2 - \sqrt{6}}{3}, - \frac{2 + \sqrt{6}}{3}$

Step 3

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

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Step 3.1

Substitute $0.14982991$ in $f (x) = x^{\frac{2}{3}} e^{x}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $0.14982991$ in the expression.

$f (0.14982991) = {(0.14982991)}^{\frac{2}{3}} e^{0.14982991}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Raise $0.14982991$ to the power of $\frac{2}{3}$ .

$f (0.14982991) = 0.28209735 e^{0.14982991}$

Step 3.1.2.2

Multiply $0.28209735$ by $e^{0.14982991}$ .

$f (0.14982991) = 0.32769463$

Step 3.1.2.3

The final answer is $0.32769463$ .

$0.32769463$

Step 3.2

The point found by substituting $0.14982991$ in $f (x) = x^{\frac{2}{3}} e^{x}$ is $(0.14982991, 0.32769463)$ . This point can be an inflection point.

$(0.14982991, 0.32769463)$

Step 3.3

Substitute $- 1.48316324$ in $f (x) = x^{\frac{2}{3}} e^{x}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 1.48316324$ in the expression.

$f (- 1.48316324) = {(- 1.48316324)}^{\frac{2}{3}} e^{- 1.48316324}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

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

$f (- 1.48316324) = {(- 1.48316324)}^{\frac{2}{3}} (\frac{1}{e^{1.48316324}})$

Step 3.3.2.2

Combine ${(- 1.48316324)}^{\frac{2}{3}}$ and $\frac{1}{e^{1.48316324}}$ .

$f (- 1.48316324) = \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}}$

Step 3.3.2.3

The final answer is $\frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}}$ .

$\frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}}$

Step 3.4

The point found by substituting $- 1.48316324$ in $f (x) = x^{\frac{2}{3}} e^{x}$ is $(- 1.48316324, \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}})$ . This point can be an inflection point.

$(- 1.48316324, \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}})$

Step 3.5

Determine the points that could be inflection points.

$(0.14982991, 0.32769463), (- 1.48316324, \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}})$

Step 4

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

$(- \infty, - 1.48316324) \cup (- 1.48316324, 0.14982991) \cup (0.14982991, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1.48316324)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.58316324$ in the expression.

$f'' (- 1.58316324) = \frac{e^{- 1.58316324} (9 {(- 1.58316324)}^{2} + 12 (- 1.58316324) - 2)}{9 {(- 1.58316324)}^{\frac{4}{3}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Move $e^{- 1.58316324}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 1.58316324) = \frac{9 {(- 1.58316324)}^{2} + 12 (- 1.58316324) - 2}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Raise $- 1.58316324$ to the power of $2$ .

$f'' (- 1.58316324) = \frac{9 \cdot 2.50640586 + 12 (- 1.58316324) - 2}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.2.2

Multiply $9$ by $2.50640586$ .

$f'' (- 1.58316324) = \frac{22.55765281 + 12 (- 1.58316324) - 2}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.2.3

Multiply $12$ by $- 1.58316324$ .

$f'' (- 1.58316324) = \frac{22.55765281 - 18.99795897 - 2}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.2.4

Subtract $18.99795897$ from $22.55765281$ .

$f'' (- 1.58316324) = \frac{3.55969384 - 2}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.2.5

Subtract $2$ from $3.55969384$ .

$f'' (- 1.58316324) = \frac{1.55969384}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.2.3

The final answer is $\frac{1.55969384}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$ .

$\frac{1.55969384}{9 {(- 1.58316324)}^{\frac{4}{3}} e^{1.58316324}}$

Step 5.3

At $- 1.58316324$ , the second derivative is $0.01928428$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 1.48316324)$ .

Increasing on $(- \infty, - 1.48316324)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 1.48316324, 0.14982991)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.66666666$ in the expression.

$f'' (- 0.66666666) = \frac{e^{- 0.66666666} (9 {(- 0.66666666)}^{2} + 12 (- 0.66666666) - 2)}{9 {(- 0.66666666)}^{\frac{4}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Move $e^{- 0.66666666}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (- 0.66666666) = \frac{9 {(- 0.66666666)}^{2} + 12 (- 0.66666666) - 2}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Raise $- 0.66666666$ to the power of $2$ .

$f'' (- 0.66666666) = \frac{9 \cdot 0.\overline{4} + 12 (- 0.66666666) - 2}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.2.2

Multiply $9$ by $0.\overline{4}$ .

$f'' (- 0.66666666) = \frac{4 + 12 (- 0.66666666) - 2}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.2.3

Multiply $12$ by $- 0.66666666$ .

$f'' (- 0.66666666) = \frac{4 - 8 - 2}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.2.4

Subtract $8$ from $4$ .

$f'' (- 0.66666666) = \frac{- 4 - 2}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.2.5

Subtract $2$ from $- 4$ .

$f'' (- 0.66666666) = \frac{- 6}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.3

Move the negative in front of the fraction.

$f'' (- 0.66666666) = - \frac{6}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.2.4

The final answer is $- \frac{6}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$ .

$- \frac{6}{9 {(- 0.66666666)}^{\frac{4}{3}} e^{0.66666666}}$

Step 6.3

At $- 0.66666666$ , the second derivative is $- 0.58771588$ . Since this is negative, the second derivative is decreasing on the interval $(- 1.48316324, 0.14982991)$

Decreasing on $(- 1.48316324, 0.14982991)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0.14982991, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.24982991$ in the expression.

$f'' (0.24982991) = \frac{e^{0.24982991} (9 {(0.24982991)}^{2} + 12 (0.24982991) - 2)}{9 {(0.24982991)}^{\frac{4}{3}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $0.24982991$ to the power of $2$ .

$f'' (0.24982991) = \frac{e^{0.24982991} (9 \cdot 0.06241498 + 12 (0.24982991) - 2)}{9 \cdot {0.24982991}^{\frac{4}{3}}}$

Step 7.2.1.2

Multiply $9$ by $0.06241498$ .

$f'' (0.24982991) = \frac{e^{0.24982991} (0.56173487 + 12 (0.24982991) - 2)}{9 \cdot {0.24982991}^{\frac{4}{3}}}$

Step 7.2.1.3

Multiply $12$ by $0.24982991$ .

$f'' (0.24982991) = \frac{e^{0.24982991} (0.56173487 + 2.99795897 - 2)}{9 \cdot {0.24982991}^{\frac{4}{3}}}$

Step 7.2.1.4

Add $0.56173487$ and $2.99795897$ .

$f'' (0.24982991) = \frac{e^{0.24982991} (3.55969384 - 2)}{9 \cdot {0.24982991}^{\frac{4}{3}}}$

Step 7.2.1.5

Subtract $2$ from $3.55969384$ .

$f'' (0.24982991) = \frac{e^{0.24982991} \cdot 1.55969384}{9 \cdot {0.24982991}^{\frac{4}{3}}}$

Step 7.2.2

Simplify by multiplying terms.

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Step 7.2.2.1

Raise $0.24982991$ to the power of $\frac{4}{3}$ .

$f'' (0.24982991) = \frac{e^{0.24982991} \cdot 1.55969384}{9 \cdot 0.15734728}$

Step 7.2.2.2

Multiply $e^{0.24982991}$ by $1.55969384$ .

$f'' (0.24982991) = \frac{2.00234594}{9 \cdot 0.15734728}$

Step 7.2.2.3

Simplify the expression.

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Step 7.2.2.3.1

Multiply $9$ by $0.15734728$ .

$f'' (0.24982991) = \frac{2.00234594}{1.41612555}$

Step 7.2.2.3.2

Divide $2.00234594$ by $1.41612555$ .

$f'' (0.24982991) = 1.41396073$

Step 7.2.3

The final answer is $1.41396073$ .

$1.41396073$

Step 7.3

At $0.24982991$ , the second derivative is $1.41396073$ . Since this is positive, the second derivative is increasing on the interval $(0.14982991, \infty)$ .

Increasing on $(0.14982991, \infty)$ since $f'' (x) > 0$

Step 8

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 $(- 1.48316324, \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}}), (0.14982991, 0.32769463)$ .

$(- 1.48316324, \frac{{(- 1.48316324)}^{\frac{2}{3}}}{e^{1.48316324}}), (0.14982991, 0.32769463)$

Step 9