Calculus Examples

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

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

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.2.5.1

Cancel the common factor.

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

Step 1.1.2.5.2

Divide $x^{2}$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Combine the numerators over the common denominator.

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

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.3.6.2

Subtract $3$ from $2$ .

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

Step 1.1.3.7

Move the negative in front of the fraction.

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

Step 1.1.3.8

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

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

Step 1.1.3.9

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

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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 1.2.2.3.1

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

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

Step 1.2.2.3.2

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

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

Step 1.2.2.3.3

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

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

Step 1.2.2.4

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

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

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

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

Step 1.2.2.5.2

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

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

Step 1.2.2.5.3

Move the negative in front of the fraction.

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

Step 1.2.2.6

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

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

Step 1.2.2.7

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

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

Step 1.2.2.8

Combine the numerators over the common denominator.

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

Step 1.2.2.9

Simplify the numerator.

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Step 1.2.2.9.1

Multiply $- 1$ by $3$ .

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

Step 1.2.2.9.2

Subtract $3$ from $1$ .

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

Step 1.2.2.10

Move the negative in front of the fraction.

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

Step 1.2.2.11

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

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

Step 1.2.2.12

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

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

Step 1.2.2.13

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

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Step 1.2.2.13.1

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

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

Step 1.2.2.13.2

Combine the numerators over the common denominator.

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

Step 1.2.2.13.3

Subtract $2$ from $- 2$ .

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

Step 1.2.2.13.4

Move the negative in front of the fraction.

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

Step 1.2.2.14

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

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

Step 1.2.2.15

Multiply $- 1$ by $- 1$ .

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

Step 1.2.2.16

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

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

Step 1.2.2.17

Multiply $\frac{2}{3}$ by $\frac{1}{3 x^{\frac{4}{3}}}$ .

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

Step 1.2.2.18

Multiply $3$ by $3$ .

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

Step 1.3

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

$2 x + \frac{2}{9 x^{\frac{4}{3}}}$

Step 2

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

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Step 2.1

Set the second derivative equal to $0$ .

$2 x + \frac{2}{9 x^{\frac{4}{3}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$1, 9 x^{\frac{4}{3}}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$9 x^{\frac{4}{3}}$

Step 2.3

Multiply each term in $2 x + \frac{2}{9 x^{\frac{4}{3}}} = 0$ by $9 x^{\frac{4}{3}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $2 x + \frac{2}{9 x^{\frac{4}{3}}} = 0$ by $9 x^{\frac{4}{3}}$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.3.2.1.2

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

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Step 2.3.2.1.2.1

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

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

Step 2.3.2.1.2.2

Multiply $x^{\frac{4}{3}}$ by $x$ .

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Step 2.3.2.1.2.2.1

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

$2 \cdot 9 (x^{\frac{4}{3}} x^{1}) + \frac{2}{9 x^{\frac{4}{3}}} (9 x^{\frac{4}{3}}) = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.2.2.2

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

$2 \cdot 9 x^{\frac{4}{3} + 1} + \frac{2}{9 x^{\frac{4}{3}}} (9 x^{\frac{4}{3}}) = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.2.3

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

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

Step 2.3.2.1.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.1.2.5

Add $4$ and $3$ .

$2 \cdot 9 x^{\frac{7}{3}} + \frac{2}{9 x^{\frac{4}{3}}} (9 x^{\frac{4}{3}}) = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.3

Multiply $2$ by $9$ .

$18 x^{\frac{7}{3}} + \frac{2}{9 x^{\frac{4}{3}}} (9 x^{\frac{4}{3}}) = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.4

Rewrite using the commutative property of multiplication.

$18 x^{\frac{7}{3}} + 9 \frac{2}{9 x^{\frac{4}{3}}} x^{\frac{4}{3}} = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.5

Cancel the common factor of $9$ .

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Step 2.3.2.1.5.1

Cancel the common factor.

$18 x^{\frac{7}{3}} + 9 \frac{2}{9 x^{\frac{4}{3}}} x^{\frac{4}{3}} = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.5.2

Rewrite the expression.

$18 x^{\frac{7}{3}} + \frac{2}{x^{\frac{4}{3}}} x^{\frac{4}{3}} = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.6

Cancel the common factor of $x^{\frac{4}{3}}$ .

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Step 2.3.2.1.6.1

Cancel the common factor.

$18 x^{\frac{7}{3}} + \frac{2}{x^{\frac{4}{3}}} x^{\frac{4}{3}} = 0 (9 x^{\frac{4}{3}})$

Step 2.3.2.1.6.2

Rewrite the expression.

$18 x^{\frac{7}{3}} + 2 = 0 (9 x^{\frac{4}{3}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (9 x^{\frac{4}{3}})$ .

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Step 2.3.3.1.1

Multiply $9$ by $0$ .

$18 x^{\frac{7}{3}} + 2 = 0 x^{\frac{4}{3}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{4}{3}}$ .

$18 x^{\frac{7}{3}} + 2 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $2$ from both sides of the equation.

$18 x^{\frac{7}{3}} = - 2$

Step 2.4.2

Raise each side of the equation to the power of $\frac{3}{7}$ to eliminate the fractional exponent on the left side.

${(18 x^{\frac{7}{3}})}^{\frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3

Simplify the left side.

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Step 2.4.3.1

Simplify ${(18 x^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 2.4.3.1.1

Apply the product rule to $18 x^{\frac{7}{3}}$ .

$18^{\frac{3}{7}} {(x^{\frac{7}{3}})}^{\frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.2

Multiply the exponents in ${(x^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 2.4.3.1.2.1

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

$18^{\frac{3}{7}} x^{\frac{7}{3} \cdot \frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.2.2

Cancel the common factor of $7$ .

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Step 2.4.3.1.2.2.1

Cancel the common factor.

$18^{\frac{3}{7}} x^{\frac{7}{3} \cdot \frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.2.2.2

Rewrite the expression.

$18^{\frac{3}{7}} x^{\frac{1}{3} \cdot 3} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.2.3

Cancel the common factor of $3$ .

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Step 2.4.3.1.2.3.1

Cancel the common factor.

$18^{\frac{3}{7}} x^{\frac{1}{3} \cdot 3} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.2.3.2

Rewrite the expression.

$18^{\frac{3}{7}} x^{1} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.3

Simplify.

$18^{\frac{3}{7}} x = {(- 2)}^{\frac{3}{7}}$

Step 2.4.3.1.4

Reorder factors in $18^{\frac{3}{7}} x$ .

$x \cdot 18^{\frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$

Step 2.4.4

Divide each term in $x \cdot 18^{\frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$ by $18^{\frac{3}{7}}$ and simplify.

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Step 2.4.4.1

Divide each term in $x \cdot 18^{\frac{3}{7}} = {(- 2)}^{\frac{3}{7}}$ by $18^{\frac{3}{7}}$ .

$\frac{x \cdot 18^{\frac{3}{7}}}{18^{\frac{3}{7}}} = \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor.

$\frac{x \cdot 18^{\frac{3}{7}}}{18^{\frac{3}{7}}} = \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$

Step 2.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$

Step 3

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

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Step 3.1

Substitute $\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$ in $f (x) = \frac{1}{3} x^{3} - x^{\frac{2}{3}}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$ in the expression.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{1}{3} \cdot {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{3} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to $\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{1}{3} \cdot \frac{{({(- 2)}^{\frac{3}{7}})}^{3}}{{(18^{\frac{3}{7}})}^{3}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.2

Combine.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{1 {({(- 2)}^{\frac{3}{7}})}^{3}}{3 {(18^{\frac{3}{7}})}^{3}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.3

Multiply ${({(- 2)}^{\frac{3}{7}})}^{3}$ by $1$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{({(- 2)}^{\frac{3}{7}})}^{3}}{3 {(18^{\frac{3}{7}})}^{3}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.4

Multiply the exponents in ${(18^{\frac{3}{7}})}^{3}$ .

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Step 3.1.2.1.4.1

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

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{({(- 2)}^{\frac{3}{7}})}^{3}}{3 \cdot 18^{\frac{3}{7} \cdot 3}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.4.2

Multiply $\frac{3}{7} \cdot 3$ .

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Step 3.1.2.1.4.2.1

Combine $\frac{3}{7}$ and $3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{({(- 2)}^{\frac{3}{7}})}^{3}}{3 \cdot 18^{\frac{3 \cdot 3}{7}}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.4.2.2

Multiply $3$ by $3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{({(- 2)}^{\frac{3}{7}})}^{3}}{3 \cdot 18^{\frac{9}{7}}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.5

Multiply the exponents in ${({(- 2)}^{\frac{3}{7}})}^{3}$ .

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Step 3.1.2.1.5.1

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

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{3}{7} \cdot 3}}{3 \cdot 18^{\frac{9}{7}}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.5.2

Multiply $\frac{3}{7} \cdot 3$ .

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Step 3.1.2.1.5.2.1

Combine $\frac{3}{7}$ and $3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{3 \cdot 3}{7}}}{3 \cdot 18^{\frac{9}{7}}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.5.2.2

Multiply $3$ by $3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - {(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 3.1.2.1.6

Apply the product rule to $\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{({(- 2)}^{\frac{3}{7}})}^{\frac{2}{3}}}{{(18^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 3.1.2.1.7

Multiply the exponents in ${({(- 2)}^{\frac{3}{7}})}^{\frac{2}{3}}$ .

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Step 3.1.2.1.7.1

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

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{3}{7} \cdot \frac{2}{3}}}{{(18^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 3.1.2.1.7.2

Cancel the common factor of $3$ .

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Step 3.1.2.1.7.2.1

Cancel the common factor.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{3}{7} \cdot \frac{2}{3}}}{{(18^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 3.1.2.1.7.2.2

Rewrite the expression.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{1}{7} \cdot 2}}{{(18^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 3.1.2.1.7.3

Combine $\frac{1}{7}$ and $2$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{{(18^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 3.1.2.1.8

Multiply the exponents in ${(18^{\frac{3}{7}})}^{\frac{2}{3}}$ .

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Step 3.1.2.1.8.1

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

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{3}{7} \cdot \frac{2}{3}}}$

Step 3.1.2.1.8.2

Cancel the common factor of $3$ .

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Step 3.1.2.1.8.2.1

Cancel the common factor.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{3}{7} \cdot \frac{2}{3}}}$

Step 3.1.2.1.8.2.2

Rewrite the expression.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{1}{7} \cdot 2}}$

Step 3.1.2.1.8.3

Combine $\frac{1}{7}$ and $2$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{2}{7}}}$

Step 3.1.2.2

To write $- \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{2}{7}}}$ as a fraction with a common denominator, multiply by $\frac{3 \cdot 18^{\frac{7}{7}}}{3 \cdot 18^{\frac{7}{7}}}$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{2}{7}}} \cdot \frac{3 \cdot 18^{\frac{7}{7}}}{3 \cdot 18^{\frac{7}{7}}}$

Step 3.1.2.3

Write each expression with a common denominator of $3 \cdot 18^{\frac{9}{7}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.2.3.1

Multiply $\frac{{(- 2)}^{\frac{2}{7}}}{18^{\frac{2}{7}}}$ by $\frac{3 \cdot 18^{\frac{7}{7}}}{3 \cdot 18^{\frac{7}{7}}}$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{18^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}$

Step 3.1.2.3.2

Multiply $18^{\frac{2}{7}}$ by $18^{\frac{7}{7}}$ by adding the exponents.

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Step 3.1.2.3.2.1

Move $18^{\frac{7}{7}}$ .

Step 3.1.2.3.2.2

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

Step 3.1.2.3.2.3

Combine the numerators over the common denominator.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{18^{\frac{7 + 2}{7}} \cdot 3}$

Step 3.1.2.3.2.4

Add $7$ and $2$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{18^{\frac{9}{7}} \cdot 3}$

Step 3.1.2.3.3

Reorder the factors of $18^{\frac{9}{7}} \cdot 3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}}}{3 \cdot 18^{\frac{9}{7}}} - \frac{{(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.4

Reduce the expression by cancelling the common factors.

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Step 3.1.2.4.1

Combine the numerators over the common denominator.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - {(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.4.2

Cancel the common factor of $7$ .

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Step 3.1.2.4.2.1

Cancel the common factor.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - {(- 2)}^{\frac{2}{7}} (3 \cdot 18^{\frac{7}{7}})}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.4.2.2

Rewrite the expression.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - {(- 2)}^{\frac{2}{7}} (3 \cdot 18)}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $3$ by $- 1$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - 3 {(- 2)}^{\frac{2}{7}} \cdot 18}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.5.2

Evaluate the exponent.

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - 3 {(- 2)}^{\frac{2}{7}} \cdot 18}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.5.3

Multiply $18$ by $- 3$ .

$f (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) = \frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.1.2.6

The final answer is $\frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}}$ .

$\frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}}$

Step 3.2

The point found by substituting $\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}$ in $f (x) = \frac{1}{3} x^{3} - x^{\frac{2}{3}}$ is $(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}})$ . This point can be an inflection point.

$(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}})$

Step 4

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

$(- \infty, \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}) \cup (\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.48997693$ in the expression.

$f'' (- 0.48997693) = 2 (- 0.48997693) + \frac{2}{9 {(- 0.48997693)}^{\frac{4}{3}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $- 0.48997693$ .

$f'' (- 0.48997693) = - 0.97995387 + \frac{2}{9 {(- 0.48997693)}^{\frac{4}{3}}}$

Step 5.2.2

The final answer is $- 0.97995387 + \frac{2}{9 {(- 0.48997693)}^{\frac{4}{3}}}$ .

$- 0.97995387 + \frac{2}{9 {(- 0.48997693)}^{\frac{4}{3}}}$

Step 5.3

At $- 0.48997693$ , the second derivative is $- 0.40466412$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})$

Decreasing on $(- \infty, \frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.28997693$ in the expression.

$f'' (- 0.28997693) = 2 (- 0.28997693) + \frac{2}{9 {(- 0.28997693)}^{\frac{4}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $- 0.28997693$ .

$f'' (- 0.28997693) = - 0.57995387 + \frac{2}{9 {(- 0.28997693)}^{\frac{4}{3}}}$

Step 6.2.2

The final answer is $- 0.57995387 + \frac{2}{9 {(- 0.28997693)}^{\frac{4}{3}}}$ .

$- 0.57995387 + \frac{2}{9 {(- 0.28997693)}^{\frac{4}{3}}}$

Step 6.3

At $- 0.28997693$ , the second derivative is $0.57785322$ . Since this is positive, the second derivative is increasing on the interval $(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \infty)$ .

Increasing on $(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \infty)$ since $f'' (x) > 0$

Step 7

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 point in this case is $(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}})$ .

$(\frac{{(- 2)}^{\frac{3}{7}}}{18^{\frac{3}{7}}}, \frac{{(- 2)}^{\frac{9}{7}} - 54 {(- 2)}^{\frac{2}{7}}}{3 \cdot 18^{\frac{9}{7}}})$

Step 8