Calculus Examples

$3 x - 9 x^{\frac{1}{3}}$

Step 1

Write $3 x - 9 x^{\frac{1}{3}}$ as a function.

$f (x) = 3 x - 9 x^{\frac{1}{3}}$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $3 x - 9 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 2.2

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

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Step 2.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 2.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 2.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 9 x^{\frac{1}{3}}]$ .

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Step 2.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{\frac{1}{3}}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$3 - 9 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 2.3.2

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

$3 - 9 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 2.3.3

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

$3 - 9 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 2.3.4

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

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

Step 2.3.5

Combine the numerators over the common denominator.

$3 - 9 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $3$ .

$3 - 9 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 2.3.6.2

Subtract $3$ from $1$ .

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

Step 2.3.7

Move the negative in front of the fraction.

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

Step 2.3.8

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

$3 - 9 \frac{x^{- \frac{2}{3}}}{3}$

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

Factor $3$ out of $- 9$ .

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

Step 2.3.12

Cancel the common factors.

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Step 2.3.12.1

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

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

Step 2.3.12.2

Cancel the common factor.

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

Step 2.3.12.3

Rewrite the expression.

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

Step 2.3.13

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

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

Step 3.2.3

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

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 3.2.5.2.1

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

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

Step 3.2.5.2.2

Multiply $2$ by $- 2$ .

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

Step 3.2.5.3

Move the negative in front of the fraction.

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Combine the numerators over the common denominator.

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

Step 3.2.9

Simplify the numerator.

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Step 3.2.9.1

Multiply $- 1$ by $3$ .

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

Step 3.2.9.2

Subtract $3$ from $2$ .

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

Step 3.2.10

Move the negative in front of the fraction.

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

Step 3.2.11

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

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

Step 3.2.12

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

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

Step 3.2.13

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 3.2.13.1

Move $x^{- \frac{4}{3}}$ .

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

Step 3.2.13.2

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

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

Step 3.2.13.3

Combine the numerators over the common denominator.

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

Step 3.2.13.4

Subtract $1$ from $- 4$ .

$f'' (x) = 0 - 3 (- \frac{2 x^{\frac{- 5}{3}}}{3})$

Step 3.2.13.5

Move the negative in front of the fraction.

$f'' (x) = 0 - 3 (- \frac{2 x^{- \frac{5}{3}}}{3})$

Step 3.2.14

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 - 3 (- \frac{2}{3 x^{\frac{5}{3}}})$

Step 3.2.15

Multiply $- 1$ by $- 3$ .

$f'' (x) = 0 + 3 (\frac{2}{3 x^{\frac{5}{3}}})$

Step 3.2.16

Combine $3$ and $\frac{2}{3 x^{\frac{5}{3}}}$ .

$f'' (x) = 0 + \frac{3 \cdot 2}{3 x^{\frac{5}{3}}}$

Step 3.2.17

Multiply $3$ by $2$ .

$f'' (x) = 0 + \frac{6}{3 x^{\frac{5}{3}}}$

Step 3.2.18

Factor $3$ out of $6$ .

$f'' (x) = 0 + \frac{3 \cdot 2}{3 x^{\frac{5}{3}}}$

Step 3.2.19

Cancel the common factors.

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Step 3.2.19.1

Factor $3$ out of $3 x^{\frac{5}{3}}$ .

$f'' (x) = 0 + \frac{3 \cdot 2}{3 (x^{\frac{5}{3}})}$

Step 3.2.19.2

Cancel the common factor.

$f'' (x) = 0 + \frac{3 \cdot 2}{3 x^{\frac{5}{3}}}$

Step 3.2.19.3

Rewrite the expression.

$f'' (x) = 0 + \frac{2}{x^{\frac{5}{3}}}$

Step 3.3

Add $0$ and $\frac{2}{x^{\frac{5}{3}}}$ .

$f'' (x) = \frac{2}{x^{\frac{5}{3}}}$

Step 4

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

$3 - \frac{3}{x^{\frac{2}{3}}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $3 x - 9 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 5.1.2

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

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Step 5.1.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 5.1.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 5.1.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 9 x^{\frac{1}{3}}]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 9 x^{\frac{1}{3}}]$ .

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Step 5.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{\frac{1}{3}}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$3 - 9 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 5.1.3.2

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

$3 - 9 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 5.1.3.3

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

$3 - 9 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 5.1.3.4

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

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

Step 5.1.3.5

Combine the numerators over the common denominator.

$3 - 9 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 5.1.3.6

Simplify the numerator.

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Step 5.1.3.6.1

Multiply $- 1$ by $3$ .

$3 - 9 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 5.1.3.6.2

Subtract $3$ from $1$ .

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

Step 5.1.3.7

Move the negative in front of the fraction.

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

Step 5.1.3.8

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

$3 - 9 \frac{x^{- \frac{2}{3}}}{3}$

Step 5.1.3.9

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

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

Step 5.1.3.10

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

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

Step 5.1.3.11

Factor $3$ out of $- 9$ .

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

Step 5.1.3.12

Cancel the common factors.

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Step 5.1.3.12.1

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

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

Step 5.1.3.12.2

Cancel the common factor.

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

Step 5.1.3.12.3

Rewrite the expression.

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

Step 5.1.3.13

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

$3 - \frac{3}{x^{\frac{2}{3}}} = 0$

Step 6.2

Subtract $3$ from both sides of the equation.

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

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

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

Step 6.3.2

The LCM of one and any expression is the expression.

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x^{\frac{2}{3}}$ .

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Step 6.4.2.1.1

Move the leading negative in $- \frac{3}{x^{\frac{2}{3}}}$ into the numerator.

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

Step 6.4.2.1.2

Cancel the common factor.

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

Step 6.4.2.1.3

Rewrite the expression.

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

Step 6.5

Solve the equation.

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Step 6.5.1

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

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

Step 6.5.2

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

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Step 6.5.2.1

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

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

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor.

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

Step 6.5.2.2.2

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

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

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 3$ by $- 3$ .

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

Step 6.5.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 6.5.4

Simplify the exponent.

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Step 6.5.4.1

Simplify the left side.

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Step 6.5.4.1.1

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 6.5.4.1.1.1

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

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Step 6.5.4.1.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 6.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.5.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 6.5.4.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 6.5.4.1.1.1.3

Cancel the common factor of $3$ .

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Step 6.5.4.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 6.5.4.1.1.1.3.2

Rewrite the expression.

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

Step 6.5.4.1.1.2

Simplify.

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

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

One to any power is one.

$x = \pm 1$

Step 6.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 6.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 6.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

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

Step 7.2

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

$\sqrt[3]{x^{2}} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

${\sqrt[3]{x^{2}}}^{3} = 0^{3}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

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

$x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.2.2

Rewrite the expression.

$x^{2} = 0^{3}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$x^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

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

$x = \pm \sqrt{0}$

Step 7.3.3.2

Simplify $\pm \sqrt{0}$ .

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Step 7.3.3.2.1

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

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

Step 7.3.3.2.2

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

$x = \pm 0$

Step 7.3.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = 1, - 1, 0$

Step 9

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.

$\frac{2}{{(1)}^{\frac{5}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

One to any power is one.

$\frac{2}{1}$

Step 10.2

Divide $2$ by $1$ .

$2$

Step 11

$x = 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 1$ is a local minimum

Step 12

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

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Step 12.1

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

$f (1) = 3 (1) - 9 {(1)}^{\frac{1}{3}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $3$ by $1$ .

$f (1) = 3 - 9 {(1)}^{\frac{1}{3}}$

Step 12.2.1.2

One to any power is one.

$f (1) = 3 - 9 \cdot 1$

Step 12.2.1.3

Multiply $- 9$ by $1$ .

$f (1) = 3 - 9$

Step 12.2.2

Subtract $9$ from $3$ .

$f (1) = - 6$

Step 12.2.3

The final answer is $- 6$ .

$y = - 6$

Step 13

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.

$\frac{2}{{(- 1)}^{\frac{5}{3}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the denominator.

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Step 14.1.1

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

$\frac{2}{{({(- 1)}^{3})}^{\frac{5}{3}}}$

Step 14.1.2

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

$\frac{2}{{(- 1)}^{3 (\frac{5}{3})}}$

Step 14.1.3

Cancel the common factor of $3$ .

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Step 14.1.3.1

Cancel the common factor.

$\frac{2}{{(- 1)}^{3 (\frac{5}{3})}}$

Step 14.1.3.2

Rewrite the expression.

$\frac{2}{{(- 1)}^{5}}$

Step 14.1.4

Raise $- 1$ to the power of $5$ .

$\frac{2}{- 1}$

Step 14.2

Divide $2$ by $- 1$ .

$- 2$

Step 15

$x = - 1$ 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 16

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

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Step 16.1

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

$f (- 1) = 3 (- 1) - 9 {(- 1)}^{\frac{1}{3}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 - 9 {(- 1)}^{\frac{1}{3}}$

Step 16.2.1.2

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

$f (- 1) = - 3 - 9 {({(- 1)}^{3})}^{\frac{1}{3}}$

Step 16.2.1.3

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

$f (- 1) = - 3 - 9 {(- 1)}^{3 (\frac{1}{3})}$

Step 16.2.1.4

Cancel the common factor of $3$ .

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Step 16.2.1.4.1

Cancel the common factor.

$f (- 1) = - 3 - 9 {(- 1)}^{3 (\frac{1}{3})}$

Step 16.2.1.4.2

Rewrite the expression.

$f (- 1) = - 3 - 9 \cdot - 1$

Step 16.2.1.5

Evaluate the exponent.

$f (- 1) = - 3 - 9 \cdot - 1$

Step 16.2.1.6

Multiply $- 9$ by $- 1$ .

$f (- 1) = - 3 + 9$

Step 16.2.2

Add $- 3$ and $9$ .

$f (- 1) = 6$

Step 16.2.3

The final answer is $6$ .

$y = 6$

Step 17

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.

$\frac{2}{{(0)}^{\frac{5}{3}}}$

Step 18

Evaluate the second derivative.

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Step 18.1

Simplify the expression.

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Step 18.1.1

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

$\frac{2}{{(0^{3})}^{\frac{5}{3}}}$

Step 18.1.2

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

$\frac{2}{0^{3 (\frac{5}{3})}}$

Step 18.2

Cancel the common factor of $3$ .

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Step 18.2.1

Cancel the common factor.

$\frac{2}{0^{3 (\frac{5}{3})}}$

Step 18.2.2

Rewrite the expression.

$\frac{2}{0^{5}}$

Step 18.3

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

$\frac{2}{0}$

Step 18.4

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

Undefined

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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Step 19.2.1

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

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

Step 19.2.2

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

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

Step 19.3

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

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Step 19.3.1

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

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

Step 19.3.2

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

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

Step 19.4

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

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Step 19.4.1

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

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

Step 19.4.2

Simplify the result.

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Step 19.4.2.1

Simplify each term.

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Step 19.4.2.1.1

Raise $0.5$ to the power of $\frac{2}{3}$ .

$f' (0.5) = 3 - \frac{3}{0.62996052}$

Step 19.4.2.1.2

Divide $3$ by $0.62996052$ .

$f' (0.5) = 3 - 1 \cdot 4.76220315$

Step 19.4.2.1.3

Multiply $- 1$ by $4.76220315$ .

$f' (0.5) = 3 - 4.76220315$

Step 19.4.2.2

Subtract $4.76220315$ from $3$ .

$f' (0.5) = - 1.76220315$

Step 19.4.2.3

The final answer is $- 1.76220315$ .

$- 1.76220315$

Step 19.5

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

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Step 19.5.1

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

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

Step 19.5.2

Simplify the result.

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Step 19.5.2.1

Remove parentheses.

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

Step 19.5.2.2

The final answer is $3 - \frac{3}{4^{\frac{2}{3}}}$ .

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

Step 19.6

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 19.7

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 19.8

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

$x = 1$ is a local minimum

Step 19.9

These are the local extrema for $f (x) = 3 x - 9 x^{\frac{1}{3}}$ .

$x = - 1$ is a local maximum

$x = 1$ is a local minimum

$x = - 1$ is a local maximum

$x = 1$ is a local minimum

Step 20