Calculus Examples

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

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.1.2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x^{\frac{1}{3}} (2 x - 2 + 0) + (x^{2} - 2 x + 1) \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.1.2.7

Add $2 x - 2$ and $0$ .

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

Step 1.1.2.8

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.6.2

Subtract $3$ from $1$ .

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

Step 1.1.7

Move the negative in front of the fraction.

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

Simplify.

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Step 1.1.10.1

Apply the distributive property.

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

Step 1.1.10.2

Apply the distributive property.

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

Step 1.1.10.3

Combine terms.

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Step 1.1.10.3.1

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

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Step 1.1.10.3.1.1

Move $x$ .

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

Step 1.1.10.3.1.2

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

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Step 1.1.10.3.1.2.1

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

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

Step 1.1.10.3.1.2.2

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

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

Step 1.1.10.3.1.3

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

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

Step 1.1.10.3.1.4

Combine the numerators over the common denominator.

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

Step 1.1.10.3.1.5

Add $3$ and $1$ .

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

Step 1.1.10.3.2

Move $2$ to the left of $x^{\frac{4}{3}}$ .

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

Step 1.1.10.3.3

Move $- 2$ to the left of $x^{\frac{1}{3}}$ .

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

Step 1.1.10.3.4

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

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

Step 1.1.10.3.5

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

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

Step 1.1.10.3.6

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

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Step 1.1.10.3.6.1

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

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

Step 1.1.10.3.6.2

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

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

Step 1.1.10.3.6.3

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

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

Step 1.1.10.3.6.4

Combine the numerators over the common denominator.

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

Step 1.1.10.3.6.5

Simplify the numerator.

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Step 1.1.10.3.6.5.1

Multiply $2$ by $3$ .

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

Step 1.1.10.3.6.5.2

Subtract $2$ from $6$ .

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

Step 1.1.10.3.7

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

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

Step 1.1.10.3.8

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

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

Step 1.1.10.3.9

Move $- 2$ to the left of $x$ .

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

Step 1.1.10.3.10

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

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

Step 1.1.10.3.11

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

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Step 1.1.10.3.11.1

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

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

Step 1.1.10.3.11.2

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

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Step 1.1.10.3.11.2.1

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

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

Step 1.1.10.3.11.2.2

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

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

Step 1.1.10.3.11.3

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

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

Step 1.1.10.3.11.4

Combine the numerators over the common denominator.

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

Step 1.1.10.3.11.5

Add $- 2$ and $3$ .

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

Step 1.1.10.3.12

Move the negative in front of the fraction.

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

Step 1.1.10.3.13

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

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

Step 1.1.10.3.14

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

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

Step 1.1.10.3.15

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

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

Step 1.1.10.3.16

Combine the numerators over the common denominator.

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

Step 1.1.10.3.17

Multiply $3$ by $2$ .

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

Step 1.1.10.3.18

Add $6 x^{\frac{4}{3}}$ and $x^{\frac{4}{3}}$ .

$- 2 x^{\frac{1}{3}} + \frac{7 x^{\frac{4}{3}}}{3} - \frac{2 x^{\frac{1}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.19

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

$- 2 x^{\frac{1}{3}} \cdot \frac{3}{3} - \frac{2 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.20

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

$\frac{- 2 x^{\frac{1}{3}} \cdot 3}{3} - \frac{2 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.21

Combine the numerators over the common denominator.

$\frac{- 2 x^{\frac{1}{3}} \cdot 3 - 2 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.22

Multiply $3$ by $- 2$ .

$\frac{- 6 x^{\frac{1}{3}} - 2 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.23

Subtract $2 x^{\frac{1}{3}}$ from $- 6 x^{\frac{1}{3}}$ .

$\frac{- 8 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.3.24

Move the negative in front of the fraction.

$- \frac{8 x^{\frac{1}{3}}}{3} + \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.1.10.4

Reorder terms.

$f' (x) = \frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}} - \frac{8 x^{\frac{1}{3}}}{3}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $\frac{7 x^{\frac{4}{3}}}{3} + \frac{1}{3 x^{\frac{2}{3}}} - \frac{8 x^{\frac{1}{3}}}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{7 x^{\frac{4}{3}}}{3}] + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$ .

$\frac{d}{d x} [\frac{7 x^{\frac{4}{3}}}{3}] + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [\frac{7 x^{\frac{4}{3}}}{3}]$ .

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Step 1.2.2.1

Since $\frac{7}{3}$ is constant with respect to $x$ , the derivative of $\frac{7 x^{\frac{4}{3}}}{3}$ with respect to $x$ is $\frac{7}{3} \frac{d}{d x} [x^{\frac{4}{3}}]$ .

$\frac{7}{3} \frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

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 $n = \frac{4}{3}$ .

$\frac{7}{3} (\frac{4}{3} x^{\frac{4}{3} - 1}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.3

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

$\frac{7}{3} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.4

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

$\frac{7}{3} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.5

Combine the numerators over the common denominator.

$\frac{7}{3} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.6

Simplify the numerator.

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Step 1.2.2.6.1

Multiply $- 1$ by $3$ .

$\frac{7}{3} (\frac{4}{3} x^{\frac{4 - 3}{3}}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.6.2

Subtract $3$ from $4$ .

$\frac{7}{3} (\frac{4}{3} x^{\frac{1}{3}}) + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.7

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

$\frac{7}{3} \cdot \frac{4 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.8

Multiply $\frac{7}{3}$ by $\frac{4 x^{\frac{1}{3}}}{3}$ .

$\frac{7 (4 x^{\frac{1}{3}})}{3 \cdot 3} + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.9

Multiply $4$ by $7$ .

$\frac{28 x^{\frac{1}{3}}}{3 \cdot 3} + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.2.10

Multiply $3$ by $3$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{d}{d x} [\frac{1}{3 x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3

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

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Step 1.2.3.1

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.2

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} \frac{d}{d x} [{(x^{\frac{2}{3}})}^{- 1}] + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.3

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 1.2.3.3.1

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.3.2

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- u^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.3.3

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- {(x^{\frac{2}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

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 $n = \frac{2}{3}$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.5

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

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Step 1.2.3.5.1

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{\frac{2}{3} \cdot - 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.5.2

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

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Step 1.2.3.5.2.1

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{\frac{2 \cdot - 2}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.5.2.2

Multiply $2$ by $- 2$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.5.3

Move the negative in front of the fraction.

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.6

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.7

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.8

Combine the numerators over the common denominator.

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.9

Simplify the numerator.

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Step 1.2.3.9.1

Multiply $- 1$ by $3$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.9.2

Subtract $3$ from $2$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.10

Move the negative in front of the fraction.

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}})) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.11

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.12

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.13

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

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Step 1.2.3.13.1

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.13.2

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.13.3

Combine the numerators over the common denominator.

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 x^{\frac{- 4 - 1}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.13.4

Subtract $1$ from $- 4$ .

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 x^{\frac{- 5}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.13.5

Move the negative in front of the fraction.

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2 x^{- \frac{5}{3}}}{3}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.14

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

$\frac{28 x^{\frac{1}{3}}}{9} + \frac{1}{3} (- \frac{2}{3 x^{\frac{5}{3}}}) + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.15

Multiply $\frac{1}{3}$ by $\frac{2}{3 x^{\frac{5}{3}}}$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{3 (3 x^{\frac{5}{3}})} + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.3.16

Multiply $3$ by $3$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} + \frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$

Step 1.2.4

Evaluate $\frac{d}{d x} [- \frac{8 x^{\frac{1}{3}}}{3}]$ .

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Step 1.2.4.1

Since $- \frac{8}{3}$ is constant with respect to $x$ , the derivative of $- \frac{8 x^{\frac{1}{3}}}{3}$ with respect to $x$ is $- \frac{8}{3} \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.2.4.2

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

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 1.2.4.3

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

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 1.2.4.4

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

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.2.4.5

Combine the numerators over the common denominator.

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 1.2.4.6

Simplify the numerator.

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Step 1.2.4.6.1

Multiply $- 1$ by $3$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 1.2.4.6.2

Subtract $3$ from $1$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 1.2.4.7

Move the negative in front of the fraction.

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} (\frac{1}{3} x^{- \frac{2}{3}})$

Step 1.2.4.8

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

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{3} \cdot \frac{x^{- \frac{2}{3}}}{3}$

Step 1.2.4.9

Multiply $\frac{x^{- \frac{2}{3}}}{3}$ by $\frac{8}{3}$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{x^{- \frac{2}{3}} \cdot 8}{3 \cdot 3}$

Step 1.2.4.10

Multiply $3$ by $3$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{x^{- \frac{2}{3}} \cdot 8}{9}$

Step 1.2.4.11

Move $8$ to the left of $x^{- \frac{2}{3}}$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8 \cdot x^{- \frac{2}{3}}}{9}$

Step 1.2.4.12

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

$f'' (x) = \frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}}$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}} = 0$

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.

$9, 9 x^{\frac{5}{3}}, 9 x^{\frac{2}{3}}, 1$

Step 2.2.2

Since $9, 9 x^{\frac{5}{3}}, 9 x^{\frac{2}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $9, 9, 9, 1$ then find LCM for the variable part $x^{\frac{5}{3}}, x^{\frac{2}{3}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $9, 9, 9, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 3$

Step 2.2.7

Multiply $3$ by $3$ .

$9$

Step 2.2.8

The LCM of $x^{\frac{5}{3}}, x^{\frac{2}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{5}{3}}$

Step 2.2.9

The LCM for $9, 9 x^{\frac{5}{3}}, 9 x^{\frac{2}{3}}, 1$ is the numeric part $9$ multiplied by the variable part.

$9 x^{\frac{5}{3}}$

Step 2.3

Multiply each term in $\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{28 x^{\frac{1}{3}}}{9} - \frac{2}{9 x^{\frac{5}{3}}} - \frac{8}{9 x^{\frac{2}{3}}} = 0$ by $9 x^{\frac{5}{3}}$ .

$\frac{28 x^{\frac{1}{3}}}{9} (9 x^{\frac{5}{3}}) - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

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.

$9 \frac{28 x^{\frac{1}{3}}}{9} x^{\frac{5}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.2

Cancel the common factor of $9$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$9 \frac{28 x^{\frac{1}{3}}}{9} x^{\frac{5}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.2.2

Rewrite the expression.

$28 x^{\frac{1}{3}} x^{\frac{5}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.3

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{5}{3}}$ by adding the exponents.

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Step 2.3.2.1.3.1

Move $x^{\frac{5}{3}}$ .

$28 (x^{\frac{5}{3}} x^{\frac{1}{3}}) - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.3.2

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

$28 x^{\frac{5}{3} + \frac{1}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.3.3

Combine the numerators over the common denominator.

$28 x^{\frac{5 + 1}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.3.4

Add $5$ and $1$ .

$28 x^{\frac{6}{3}} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.3.5

Divide $6$ by $3$ .

$28 x^{2} - \frac{2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.4

Cancel the common factor of $9 x^{\frac{5}{3}}$ .

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Step 2.3.2.1.4.1

Move the leading negative in $- \frac{2}{9 x^{\frac{5}{3}}}$ into the numerator.

$28 x^{2} + \frac{- 2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.4.2

Cancel the common factor.

$28 x^{2} + \frac{- 2}{9 x^{\frac{5}{3}}} (9 x^{\frac{5}{3}}) - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.4.3

Rewrite the expression.

$28 x^{2} - 2 - \frac{8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.5

Cancel the common factor of $x^{\frac{2}{3}} \cdot 9$ .

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Step 2.3.2.1.5.1

Move the leading negative in $- \frac{8}{9 x^{\frac{2}{3}}}$ into the numerator.

$28 x^{2} - 2 + \frac{- 8}{9 x^{\frac{2}{3}}} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.5.2

Factor $x^{\frac{2}{3}} \cdot 9$ out of $9 x^{\frac{2}{3}}$ .

$28 x^{2} - 2 + \frac{- 8}{x^{\frac{2}{3}} \cdot 9 (1)} (9 x^{\frac{5}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.5.3

Factor $x^{\frac{2}{3}} \cdot 9$ out of $9 x^{\frac{5}{3}}$ .

$28 x^{2} - 2 + \frac{- 8}{x^{\frac{2}{3}} \cdot 9 (1)} (x^{\frac{2}{3}} \cdot 9 (x^{\frac{3}{3}})) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.5.4

Cancel the common factor.

$28 x^{2} - 2 + \frac{- 8}{x^{\frac{2}{3}} \cdot 9 \cdot 1} (x^{\frac{2}{3}} \cdot 9 x^{\frac{3}{3}}) = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.5.5

Rewrite the expression.

$28 x^{2} - 2 - 8 x^{\frac{3}{3}} = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.6

Divide $3$ by $3$ .

$28 x^{2} - 2 - 8 x^{1} = 0 (9 x^{\frac{5}{3}})$

Step 2.3.2.1.7

Simplify.

$28 x^{2} - 2 - 8 x = 0 (9 x^{\frac{5}{3}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (9 x^{\frac{5}{3}})$ .

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Step 2.3.3.1.1

Multiply $9$ by $0$ .

$28 x^{2} - 2 - 8 x = 0 x^{\frac{5}{3}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{5}{3}}$ .

$28 x^{2} - 2 - 8 x = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Factor the left side of the equation.

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Step 2.4.1.1

Factor $2$ out of $28 x^{2} - 2 - 8 x$ .

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Step 2.4.1.1.1

Factor $2$ out of $28 x^{2}$ .

$2 (14 x^{2}) - 2 - 8 x = 0$

Step 2.4.1.1.2

Factor $2$ out of $- 2$ .

$2 (14 x^{2}) + 2 (- 1) - 8 x = 0$

Step 2.4.1.1.3

Factor $2$ out of $- 8 x$ .

$2 (14 x^{2}) + 2 (- 1) + 2 (- 4 x) = 0$

Step 2.4.1.1.4

Factor $2$ out of $2 (14 x^{2}) + 2 (- 1)$ .

$2 (14 x^{2} - 1) + 2 (- 4 x) = 0$

Step 2.4.1.1.5

Factor $2$ out of $2 (14 x^{2} - 1) + 2 (- 4 x)$ .

$2 (14 x^{2} - 1 - 4 x) = 0$

Step 2.4.1.2

Reorder terms.

$2 (14 x^{2} - 4 x - 1) = 0$

Step 2.4.2

Divide each term in $2 (14 x^{2} - 4 x - 1) = 0$ by $2$ and simplify.

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Step 2.4.2.1

Divide each term in $2 (14 x^{2} - 4 x - 1) = 0$ by $2$ .

$\frac{2 (14 x^{2} - 4 x - 1)}{2} = \frac{0}{2}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{2 (14 x^{2} - 4 x - 1)}{2} = \frac{0}{2}$

Step 2.4.2.2.1.2

Divide $14 x^{2} - 4 x - 1$ by $1$ .

$14 x^{2} - 4 x - 1 = \frac{0}{2}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $0$ by $2$ .

$14 x^{2} - 4 x - 1 = 0$

Step 2.4.3

Use the quadratic formula to find the solutions.

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

Step 2.4.4

Substitute the values $a = 14$ , $b = - 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (14 \cdot - 1)}}{2 \cdot 14}$

Step 2.4.5

Simplify.

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Step 2.4.5.1

Simplify the numerator.

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Step 2.4.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 14 \cdot - 1}}{2 \cdot 14}$

Step 2.4.5.1.2

Multiply $- 4 \cdot 14 \cdot - 1$ .

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Step 2.4.5.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{4 \pm \sqrt{16 - 56 \cdot - 1}}{2 \cdot 14}$

Step 2.4.5.1.2.2

Multiply $- 56$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 56}}{2 \cdot 14}$

Step 2.4.5.1.3

Add $16$ and $56$ .

$x = \frac{4 \pm \sqrt{72}}{2 \cdot 14}$

Step 2.4.5.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 2.4.5.1.4.1

Factor $36$ out of $72$ .

$x = \frac{4 \pm \sqrt{36 (2)}}{2 \cdot 14}$

Step 2.4.5.1.4.2

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

$x = \frac{4 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 14}$

Step 2.4.5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 6 \sqrt{2}}{2 \cdot 14}$

Step 2.4.5.2

Multiply $2$ by $14$ .

$x = \frac{4 \pm 6 \sqrt{2}}{28}$

Step 2.4.5.3

Simplify $\frac{4 \pm 6 \sqrt{2}}{28}$ .

$x = \frac{2 \pm 3 \sqrt{2}}{14}$

Step 2.4.6

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

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Step 2.4.6.1

Simplify the numerator.

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Step 2.4.6.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 14 \cdot - 1}}{2 \cdot 14}$

Step 2.4.6.1.2

Multiply $- 4 \cdot 14 \cdot - 1$ .

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Step 2.4.6.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{4 \pm \sqrt{16 - 56 \cdot - 1}}{2 \cdot 14}$

Step 2.4.6.1.2.2

Multiply $- 56$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 56}}{2 \cdot 14}$

Step 2.4.6.1.3

Add $16$ and $56$ .

$x = \frac{4 \pm \sqrt{72}}{2 \cdot 14}$

Step 2.4.6.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 2.4.6.1.4.1

Factor $36$ out of $72$ .

$x = \frac{4 \pm \sqrt{36 (2)}}{2 \cdot 14}$

Step 2.4.6.1.4.2

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

$x = \frac{4 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 14}$

Step 2.4.6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 6 \sqrt{2}}{2 \cdot 14}$

Step 2.4.6.2

Multiply $2$ by $14$ .

$x = \frac{4 \pm 6 \sqrt{2}}{28}$

Step 2.4.6.3

Simplify $\frac{4 \pm 6 \sqrt{2}}{28}$ .

$x = \frac{2 \pm 3 \sqrt{2}}{14}$

Step 2.4.6.4

Change the $\pm$ to $+$ .

$x = \frac{2 + 3 \sqrt{2}}{14}$

Step 2.4.7

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

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Step 2.4.7.1

Simplify the numerator.

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Step 2.4.7.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 14 \cdot - 1}}{2 \cdot 14}$

Step 2.4.7.1.2

Multiply $- 4 \cdot 14 \cdot - 1$ .

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Step 2.4.7.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{4 \pm \sqrt{16 - 56 \cdot - 1}}{2 \cdot 14}$

Step 2.4.7.1.2.2

Multiply $- 56$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 56}}{2 \cdot 14}$

Step 2.4.7.1.3

Add $16$ and $56$ .

$x = \frac{4 \pm \sqrt{72}}{2 \cdot 14}$

Step 2.4.7.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 2.4.7.1.4.1

Factor $36$ out of $72$ .

$x = \frac{4 \pm \sqrt{36 (2)}}{2 \cdot 14}$

Step 2.4.7.1.4.2

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

$x = \frac{4 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 14}$

Step 2.4.7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 6 \sqrt{2}}{2 \cdot 14}$

Step 2.4.7.2

Multiply $2$ by $14$ .

$x = \frac{4 \pm 6 \sqrt{2}}{28}$

Step 2.4.7.3

Simplify $\frac{4 \pm 6 \sqrt{2}}{28}$ .

$x = \frac{2 \pm 3 \sqrt{2}}{14}$

Step 2.4.7.4

Change the $\pm$ to $-$ .

$x = \frac{2 - 3 \sqrt{2}}{14}$

Step 2.4.8

The final answer is the combination of both solutions.

$x = \frac{2 + 3 \sqrt{2}}{14}, \frac{2 - 3 \sqrt{2}}{14}$

Step 3

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

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Step 3.1

Substitute $0.4459029$ in $f (x) = x^{\frac{1}{3}} (x^{2} - 2 x + 1)$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $0.4459029$ in the expression.

$f (0.4459029) = {(0.4459029)}^{\frac{1}{3}} ({(0.4459029)}^{2} - 2 \cdot 0.4459029 + 1)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Raise $0.4459029$ to the power of $\frac{1}{3}$ .

$f (0.4459029) = 0.76397667 ({(0.4459029)}^{2} - 2 \cdot 0.4459029 + 1)$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

Raise $0.4459029$ to the power of $2$ .

$f (0.4459029) = 0.76397667 (0.1988294 - 2 \cdot 0.4459029 + 1)$

Step 3.1.2.2.2

Multiply $- 2$ by $0.4459029$ .

$f (0.4459029) = 0.76397667 (0.1988294 - 0.89180581 + 1)$

Step 3.1.2.3

Simplify the expression.

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Step 3.1.2.3.1

Subtract $0.89180581$ from $0.1988294$ .

$f (0.4459029) = 0.76397667 (- 0.69297641 + 1)$

Step 3.1.2.3.2

Add $- 0.69297641$ and $1$ .

$f (0.4459029) = 0.76397667 \cdot 0.30702358$

Step 3.1.2.3.3

Multiply $0.76397667$ by $0.30702358$ .

$f (0.4459029) = 0.23455886$

Step 3.1.2.4

The final answer is $0.23455886$ .

$0.23455886$

Step 3.2

The point found by substituting $0.4459029$ in $f (x) = x^{\frac{1}{3}} (x^{2} - 2 x + 1)$ is $(0.4459029, 0.23455886)$ . This point can be an inflection point.

$(0.4459029, 0.23455886)$

Step 3.3

Substitute $- 0.16018862$ in $f (x) = x^{\frac{1}{3}} (x^{2} - 2 x + 1)$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 0.16018862$ in the expression.

$f (- 0.16018862) = {(- 0.16018862)}^{\frac{1}{3}} ({(- 0.16018862)}^{2} - 2 \cdot - 0.16018862 + 1)$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Raise $- 0.16018862$ to the power of $2$ .

$f (- 0.16018862) = {(- 0.16018862)}^{\frac{1}{3}} (0.02566039 - 2 \cdot - 0.16018862 + 1)$

Step 3.3.2.1.2

Multiply $- 2$ by $- 0.16018862$ .

$f (- 0.16018862) = {(- 0.16018862)}^{\frac{1}{3}} (0.02566039 + 0.32037724 + 1)$

Step 3.3.2.2

Simplify the expression.

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Step 3.3.2.2.1

Add $0.02566039$ and $0.32037724$ .

$f (- 0.16018862) = {(- 0.16018862)}^{\frac{1}{3}} (0.34603763 + 1)$

Step 3.3.2.2.2

Add $0.34603763$ and $1$ .

$f (- 0.16018862) = {(- 0.16018862)}^{\frac{1}{3}} \cdot 1.34603763$

Step 3.3.2.2.3

Move $1.34603763$ to the left of ${(- 0.16018862)}^{\frac{1}{3}}$ .

$f (- 0.16018862) = 1.34603763 {(- 0.16018862)}^{\frac{1}{3}}$

Step 3.3.2.3

The final answer is $1.34603763 {(- 0.16018862)}^{\frac{1}{3}}$ .

$1.34603763 {(- 0.16018862)}^{\frac{1}{3}}$

Step 3.4

The point found by substituting $- 0.16018862$ in $f (x) = x^{\frac{1}{3}} (x^{2} - 2 x + 1)$ is $(- 0.16018862, 1.34603763 {(- 0.16018862)}^{\frac{1}{3}})$ . This point can be an inflection point.

$(- 0.16018862, 1.34603763 {(- 0.16018862)}^{\frac{1}{3}})$

Step 3.5

Determine the points that could be inflection points.

$(0.4459029, 0.23455886), (- 0.16018862, 1.34603763 {(- 0.16018862)}^{\frac{1}{3}})$

Step 4

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

$(- \infty, - 0.16018862) \cup (- 0.16018862, 0.4459029) \cup (0.4459029, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 0.16018862)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.26018862$ in the expression.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8}{9 {(- 0.26018862)}^{\frac{2}{3}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

To write $- \frac{8}{9 {(- 0.26018862)}^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(- 0.26018862)}^{\frac{3}{3}}}{{(- 0.26018862)}^{\frac{3}{3}}}$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8}{9 {(- 0.26018862)}^{\frac{2}{3}}} \cdot \frac{{(- 0.26018862)}^{\frac{3}{3}}}{{(- 0.26018862)}^{\frac{3}{3}}}$

Step 5.2.2

Write each expression with a common denominator of $9 {(- 0.26018862)}^{\frac{5}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.1

Multiply $\frac{8}{9 {(- 0.26018862)}^{\frac{2}{3}}}$ by $\frac{{(- 0.26018862)}^{\frac{3}{3}}}{{(- 0.26018862)}^{\frac{3}{3}}}$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8 {(- 0.26018862)}^{\frac{3}{3}}}{9 {(- 0.26018862)}^{\frac{2}{3}} {(- 0.26018862)}^{\frac{3}{3}}}$

Step 5.2.2.2

Multiply ${(- 0.26018862)}^{\frac{2}{3}}$ by ${(- 0.26018862)}^{\frac{3}{3}}$ by adding the exponents.

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Step 5.2.2.2.1

Move ${(- 0.26018862)}^{\frac{3}{3}}$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8 {(- 0.26018862)}^{\frac{3}{3}}}{9 ({(- 0.26018862)}^{\frac{3}{3}} {(- 0.26018862)}^{\frac{2}{3}})}$

Step 5.2.2.2.2

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

Step 5.2.2.2.3

Combine the numerators over the common denominator.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8 {(- 0.26018862)}^{\frac{3}{3}}}{9 {(- 0.26018862)}^{\frac{3 + 2}{3}}}$

Step 5.2.2.2.4

Add $3$ and $2$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(- 0.26018862)}^{\frac{5}{3}}} - \frac{8 {(- 0.26018862)}^{\frac{3}{3}}}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.3

Combine the numerators over the common denominator.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{- 2 - 8 {(- 0.26018862)}^{\frac{3}{3}}}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Cancel the common factor of $3$ .

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Step 5.2.4.1.1

Cancel the common factor.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{- 2 - 8 {(- 0.26018862)}^{\frac{3}{3}}}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.4.1.2

Rewrite the expression.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{- 2 - 8 \cdot - 0.26018862}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.4.2

Simplify the numerator.

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Step 5.2.4.2.1

Evaluate the exponent.

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{- 2 - 8 \cdot - 0.26018862}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.4.2.2

Multiply $- 8$ by $- 0.26018862$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{- 2 + 2.08150896}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.4.2.3

Add $- 2$ and $2.08150896$ .

$f'' (- 0.26018862) = \frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{0.08150896}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.2.5

The final answer is $\frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{0.08150896}{9 {(- 0.26018862)}^{\frac{5}{3}}}$ .

$\frac{28 {(- 0.26018862)}^{\frac{1}{3}}}{9} + \frac{0.08150896}{9 {(- 0.26018862)}^{\frac{5}{3}}}$

Step 5.3

At $- 0.26018862$ , the second derivative is $- 2.07155288$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 0.16018862)$

Decreasing on $(- \infty, - 0.16018862)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 0.16018862, 0.4459029)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.\overline{142857}$ in the expression.

$f'' (0.\overline{142857}) = \frac{28 {(0.\overline{142857})}^{\frac{1}{3}}}{9} - \frac{2}{9 {(0.\overline{142857})}^{\frac{5}{3}}} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $0.\overline{142857}$ to the power of $\frac{1}{3}$ .

$f'' (0.\overline{142857}) = \frac{28 \cdot 0.52275795}{9} - \frac{2}{9 {(0.\overline{142857})}^{\frac{5}{3}}} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.2

Multiply $28$ by $0.52275795$ .

$f'' (0.\overline{142857}) = \frac{14.63722284}{9} - \frac{2}{9 {(0.\overline{142857})}^{\frac{5}{3}}} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.3

Divide $14.63722284$ by $9$ .

$f'' (0.\overline{142857}) = 1.62635809 - \frac{2}{9 {(0.\overline{142857})}^{\frac{5}{3}}} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.4

Raise $0.\overline{142857}$ to the power of $\frac{5}{3}$ .

$f'' (0.\overline{142857}) = 1.62635809 - \frac{2}{9 \cdot 0.03903941} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.5

Multiply $9$ by $0.03903941$ .

$f'' (0.\overline{142857}) = 1.62635809 - \frac{2}{0.3513547} - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.6

Divide $2$ by $0.3513547$ .

$f'' (0.\overline{142857}) = 1.62635809 - 1 \cdot 5.69225332 - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.7

Multiply $- 1$ by $5.69225332$ .

$f'' (0.\overline{142857}) = 1.62635809 - 5.69225332 - \frac{8}{9 {(0.\overline{142857})}^{\frac{2}{3}}}$

Step 6.2.1.8

Raise $0.\overline{142857}$ to the power of $\frac{2}{3}$ .

$f'' (0.\overline{142857}) = 1.62635809 - 5.69225332 - \frac{8}{9 \cdot 0.27327588}$

Step 6.2.1.9

Multiply $9$ by $0.27327588$ .

$f'' (0.\overline{142857}) = 1.62635809 - 5.69225332 - \frac{8}{2.45948294}$

Step 6.2.1.10

Divide $8$ by $2.45948294$ .

$f'' (0.\overline{142857}) = 1.62635809 - 5.69225332 - 1 \cdot 3.25271618$

Step 6.2.1.11

Multiply $- 1$ by $3.25271618$ .

$f'' (0.\overline{142857}) = 1.62635809 - 5.69225332 - 3.25271618$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $5.69225332$ from $1.62635809$ .

$f'' (0.\overline{142857}) = - 4.06589523 - 3.25271618$

Step 6.2.2.2

Subtract $3.25271618$ from $- 4.06589523$ .

$f'' (0.\overline{142857}) = - 7.31861142$

Step 6.2.3

The final answer is $- 7.31861142$ .

$- 7.31861142$

Step 6.3

At $0.\overline{142857}$ , the second derivative is $- 7.31861142$ . Since this is negative, the second derivative is decreasing on the interval $(- 0.16018862, 0.4459029)$

Decreasing on $(- 0.16018862, 0.4459029)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0.4459029, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.5459029$ in the expression.

$f'' (0.5459029) = \frac{28 {(0.5459029)}^{\frac{1}{3}}}{9} - \frac{2}{9 {(0.5459029)}^{\frac{5}{3}}} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $0.5459029$ to the power of $\frac{1}{3}$ .

$f'' (0.5459029) = \frac{28 \cdot 0.81728175}{9} - \frac{2}{9 {(0.5459029)}^{\frac{5}{3}}} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.2

Multiply $28$ by $0.81728175$ .

$f'' (0.5459029) = \frac{22.88388904}{9} - \frac{2}{9 {(0.5459029)}^{\frac{5}{3}}} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.3

Divide $22.88388904$ by $9$ .

$f'' (0.5459029) = 2.54265433 - \frac{2}{9 {(0.5459029)}^{\frac{5}{3}}} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.4

Raise $0.5459029$ to the power of $\frac{5}{3}$ .

$f'' (0.5459029) = 2.54265433 - \frac{2}{9 \cdot 0.36463555} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.5

Multiply $9$ by $0.36463555$ .

$f'' (0.5459029) = 2.54265433 - \frac{2}{3.28171997} - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.6

Divide $2$ by $3.28171997$ .

$f'' (0.5459029) = 2.54265433 - 1 \cdot 0.60943652 - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.7

Multiply $- 1$ by $0.60943652$ .

$f'' (0.5459029) = 2.54265433 - 0.60943652 - \frac{8}{9 {(0.5459029)}^{\frac{2}{3}}}$

Step 7.2.1.8

Raise $0.5459029$ to the power of $\frac{2}{3}$ .

$f'' (0.5459029) = 2.54265433 - 0.60943652 - \frac{8}{9 \cdot 0.66794946}$

Step 7.2.1.9

Multiply $9$ by $0.66794946$ .

$f'' (0.5459029) = 2.54265433 - 0.60943652 - \frac{8}{6.01154515}$

Step 7.2.1.10

Divide $8$ by $6.01154515$ .

$f'' (0.5459029) = 2.54265433 - 0.60943652 - 1 \cdot 1.33077267$

Step 7.2.1.11

Multiply $- 1$ by $1.33077267$ .

$f'' (0.5459029) = 2.54265433 - 0.60943652 - 1.33077267$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $0.60943652$ from $2.54265433$ .

$f'' (0.5459029) = 1.93321781 - 1.33077267$

Step 7.2.2.2

Subtract $1.33077267$ from $1.93321781$ .

$f'' (0.5459029) = 0.60244514$

Step 7.2.3

The final answer is $0.60244514$ .

$0.60244514$

Step 7.3

At $0.5459029$ , the second derivative is $0.60244514$ . Since this is positive, the second derivative is increasing on the interval $(0.4459029, \infty)$ .

Increasing on $(0.4459029, \infty)$ 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 point in this case is $(0.4459029, 0.23455886)$ .

$(0.4459029, 0.23455886)$

Step 9