Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.6.2

Subtract $3$ from $2$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.3.10

Multiply $2$ by $3$ .

$2 + \frac{6 x^{- \frac{1}{3}}}{3}$

Step 1.3.11

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

$2 + \frac{6}{3 x^{\frac{1}{3}}}$

Step 1.3.12

Factor $3$ out of $6$ .

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

Step 1.3.13

Cancel the common factors.

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Step 1.3.13.1

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

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

Step 1.3.13.2

Cancel the common factor.

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

Step 1.3.13.3

Rewrite the expression.

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (\frac{2}{x^{\frac{1}{3}}})$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 0 + 2 \frac{d}{d x} (\frac{1}{x^{\frac{1}{3}}})$

Step 2.2.2

Rewrite $\frac{1}{x^{\frac{1}{3}}}$ as ${(x^{\frac{1}{3}})}^{- 1}$ .

$f'' (x) = 0 + 2 \frac{d}{d x} ({(x^{\frac{1}{3}})}^{- 1})$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{3}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

$f'' (x) = 0 + 2 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{3}}))$

Step 2.2.3.2

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

$f'' (x) = 0 + 2 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

$f'' (x) = 0 + 2 (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 2.2.4

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

$f'' (x) = 0 + 2 (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = 0 + 2 (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.5.2

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

$f'' (x) = 0 + 2 (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.5.3

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.2.6

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}))$

Step 2.2.7

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 2.2.8

Combine the numerators over the common denominator.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}))$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $3$ .

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}}))$

Step 2.2.9.2

Subtract $3$ from $1$ .

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}}))$

Step 2.2.10

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}}))$

Step 2.2.11

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3})$

Step 2.2.12

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

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3})$

Step 2.2.13

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

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Step 2.2.13.1

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

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3})$

Step 2.2.13.2

Combine the numerators over the common denominator.

$f'' (x) = 0 + 2 (- \frac{x^{\frac{- 2 - 2}{3}}}{3})$

Step 2.2.13.3

Subtract $2$ from $- 2$ .

$f'' (x) = 0 + 2 (- \frac{x^{\frac{- 4}{3}}}{3})$

Step 2.2.13.4

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{4}{3}}}{3})$

Step 2.2.14

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 + 2 (- \frac{1}{3 x^{\frac{4}{3}}})$

Step 2.2.15

Multiply $- 1$ by $2$ .

$f'' (x) = 0 - 2 \frac{1}{3 x^{\frac{4}{3}}}$

Step 2.2.16

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

$f'' (x) = 0 + \frac{- 2}{3 x^{\frac{4}{3}}}$

Step 2.2.17

Move the negative in front of the fraction.

$f'' (x) = 0 - \frac{2}{3 x^{\frac{4}{3}}}$

Step 2.3

Subtract $\frac{2}{3 x^{\frac{4}{3}}}$ from $0$ .

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

Step 3

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

$2 + \frac{2}{x^{\frac{1}{3}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Multiply $2$ by $1$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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

Step 4.1.3.5

Combine the numerators over the common denominator.

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

Step 4.1.3.6

Simplify the numerator.

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Step 4.1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 4.1.3.6.2

Subtract $3$ from $2$ .

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

Step 4.1.3.7

Move the negative in front of the fraction.

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

Step 4.1.3.8

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

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

Step 4.1.3.9

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

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

Step 4.1.3.10

Multiply $2$ by $3$ .

$f' (x) = 2 + \frac{6 x^{- \frac{1}{3}}}{3}$

Step 4.1.3.11

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

$f' (x) = 2 + \frac{6}{3 x^{\frac{1}{3}}}$

Step 4.1.3.12

Factor $3$ out of $6$ .

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

Step 4.1.3.13

Cancel the common factors.

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Step 4.1.3.13.1

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

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

Step 4.1.3.13.2

Cancel the common factor.

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

Step 4.1.3.13.3

Rewrite the expression.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 + \frac{2}{x^{\frac{1}{3}}}$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $2 + \frac{2}{x^{\frac{1}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 + \frac{2}{x^{\frac{1}{3}}} = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$\frac{2}{x^{\frac{1}{3}}} = - 2$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{\frac{1}{3}}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

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

Step 5.4

Multiply each term in $\frac{2}{x^{\frac{1}{3}}} = - 2$ by $x^{\frac{1}{3}}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $\frac{2}{x^{\frac{1}{3}}} = - 2$ by $x^{\frac{1}{3}}$ .

$\frac{2}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = - 2 x^{\frac{1}{3}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2}{x^{\frac{1}{3}}} x^{\frac{1}{3}} = - 2 x^{\frac{1}{3}}$

Step 5.4.2.1.2

Rewrite the expression.

$2 = - 2 x^{\frac{1}{3}}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- 2 x^{\frac{1}{3}} = 2$ .

$- 2 x^{\frac{1}{3}} = 2$

Step 5.5.2

Divide each term in $- 2 x^{\frac{1}{3}} = 2$ by $- 2$ and simplify.

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Step 5.5.2.1

Divide each term in $- 2 x^{\frac{1}{3}} = 2$ by $- 2$ .

$\frac{- 2 x^{\frac{1}{3}}}{- 2} = \frac{2}{- 2}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{- 2 x^{\frac{1}{3}}}{- 2} = \frac{2}{- 2}$

Step 5.5.2.2.2

Divide $x^{\frac{1}{3}}$ by $1$ .

$x^{\frac{1}{3}} = \frac{2}{- 2}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Divide $2$ by $- 2$ .

$x^{\frac{1}{3}} = - 1$

Step 5.5.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 5.5.4

Simplify the exponent.

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Step 5.5.4.1

Simplify the left side.

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Step 5.5.4.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 5.5.4.1.1.1

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

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Step 5.5.4.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 5.5.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.5.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 5.5.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 1)}^{3}$

Step 5.5.4.1.1.2

Simplify.

$x = {(- 1)}^{3}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

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

$x = - 1$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 6.1.2

Anything raised to $1$ is the base itself.

$2 + \frac{2}{\sqrt[3]{x}}$

Step 6.2

Set the denominator in $\frac{2}{\sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x}}^{3} = 0^{3}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 6.3.2.2.1.1

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

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Step 6.3.2.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 6.3.2.2.1.2

Simplify.

$x = 0^{3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x = 0$

Step 7

Critical points to evaluate.

$x = - 1, 0$

Step 8

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 9.1.2

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

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

Step 9.1.3

Cancel the common factor of $3$ .

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Step 9.1.3.1

Cancel the common factor.

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

Step 9.1.3.2

Rewrite the expression.

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

Step 9.1.4

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

$- \frac{2}{3 \cdot 1}$

Step 9.2

Multiply $3$ by $1$ .

$- \frac{2}{3}$

Step 10

$x = - 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 1$ is a local maximum

Step 11

Find the y-value when $x = - 1$ .

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Step 11.1

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

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

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $2$ by $- 1$ .

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

Step 11.2.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 11.2.1.3

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

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

Step 11.2.1.4

Cancel the common factor of $3$ .

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Step 11.2.1.4.1

Cancel the common factor.

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

Step 11.2.1.4.2

Rewrite the expression.

$f (- 1) = - 2 + 3 {(- 1)}^{2}$

Step 11.2.1.5

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

$f (- 1) = - 2 + 3 \cdot 1$

Step 11.2.1.6

Multiply $3$ by $1$ .

$f (- 1) = - 2 + 3$

Step 11.2.2

Add $- 2$ and $3$ .

$f (- 1) = 1$

Step 11.2.3

The final answer is $1$ .

$y = 1$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2}{3 {(0)}^{\frac{4}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Rewrite $0$ as $0^{3}$ .

$- \frac{2}{3 {(0^{3})}^{\frac{4}{3}}}$

Step 13.1.2

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

$- \frac{2}{3 \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2

Cancel the common factor of $3$ .

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Step 13.2.1

Cancel the common factor.

$- \frac{2}{3 \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2.2

Rewrite the expression.

$- \frac{2}{3 \cdot 0^{4}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$- \frac{2}{3 \cdot 0}$

Step 13.3.2

Multiply $3$ by $0$ .

$- \frac{2}{0}$

Step 13.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{2}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, \infty)$

Step 14.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 1)$ in the first derivative $2 + \frac{2}{x^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.2.1

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

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

Step 14.2.2

The final answer is $2 + \frac{2}{{(- 4)}^{\frac{1}{3}}}$ .

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

Step 14.3

Substitute any number, such as $- 0.5$ , from the interval $(- 1, 0)$ in the first derivative $2 + \frac{2}{x^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = 2 + \frac{2}{{(- 0.5)}^{\frac{1}{3}}}$

Step 14.3.2

The final answer is $2 + \frac{2}{{(- 0.5)}^{\frac{1}{3}}}$ .

$2 + \frac{2}{{(- 0.5)}^{\frac{1}{3}}}$

Step 14.4

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $2 + \frac{2}{x^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.4.1

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

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

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify each term.

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Step 14.4.2.1.1

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

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

Step 14.4.2.1.2

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

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Step 14.4.2.1.2.1

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

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Step 14.4.2.1.2.1.1

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

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

Step 14.4.2.1.2.1.2

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

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

Step 14.4.2.1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 14.4.2.1.2.3

Combine the numerators over the common denominator.

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

Step 14.4.2.1.2.4

Subtract $1$ from $3$ .

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

Step 14.4.2.2

The final answer is $2 + 2^{\frac{2}{3}}$ .

$2 + 2^{\frac{2}{3}}$

Step 14.5

Since the first derivative changed signs from positive to negative around $x = - 1$ , then $x = - 1$ is a local maximum.

$x = - 1$ is a local maximum

Step 14.6

Since the first derivative changed signs from negative to positive around $x = 0$ , then $x = 0$ is a local minimum.

$x = 0$ is a local minimum

Step 14.7

These are the local extrema for $f (x) = 2 x + 3 x^{\frac{2}{3}}$ .

$x = - 1$ is a local maximum

$x = 0$ is a local minimum

$x = - 1$ is a local maximum

$x = 0$ is a local minimum

Step 15