Calculus Examples

$y = x {(12 - x)}^{\frac{1}{3}}$

Step 1

Write $y = x {(12 - x)}^{\frac{1}{3}}$ as a function.

$f (x) = x {(12 - x)}^{\frac{1}{3}}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $f (x) = x$ and $g (x) = {(12 - x)}^{\frac{1}{3}}$ .

$x \frac{d}{d x} [{(12 - x)}^{\frac{1}{3}}] + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 12 - x$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u$ as $12 - x$ .

$x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.2.2

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

$x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.2.3

Replace all occurrences of $u$ with $12 - x$ .

$x (\frac{1}{3} {(12 - x)}^{\frac{1}{3} - 1} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.3

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

$x (\frac{1}{3} {(12 - x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.4

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

$x (\frac{1}{3} {(12 - x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.5

Combine the numerators over the common denominator.

$x (\frac{1}{3} {(12 - x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $- 1$ by $3$ .

$x (\frac{1}{3} {(12 - x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.6.2

Subtract $3$ from $1$ .

$x (\frac{1}{3} {(12 - x)}^{\frac{- 2}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.7

Combine fractions.

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Step 2.1.7.1

Move the negative in front of the fraction.

$x (\frac{1}{3} {(12 - x)}^{- \frac{2}{3}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.7.2

Combine $\frac{1}{3}$ and ${(12 - x)}^{- \frac{2}{3}}$ .

$x (\frac{{(12 - x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.7.3

Move ${(12 - x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{3 {(12 - x)}^{\frac{2}{3}}} \frac{d}{d x} [12 - x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.7.4

Combine $\frac{1}{3 {(12 - x)}^{\frac{2}{3}}}$ and $x$ .

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} \frac{d}{d x} [12 - x] + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.8

By the Sum Rule, the derivative of $12 - x$ with respect to $x$ is $\frac{d}{d x} [12] + \frac{d}{d x} [- x]$ .

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} (\frac{d}{d x} [12] + \frac{d}{d x} [- x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.9

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} (0 + \frac{d}{d x} [- x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.10

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} \frac{d}{d x} [- x] + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.11

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

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} (- \frac{d}{d x} [x]) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.12

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

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} (- 1 \cdot 1) + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.13

Combine fractions.

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Step 2.1.13.1

Multiply $- 1$ by $1$ .

$\frac{x}{3 {(12 - x)}^{\frac{2}{3}}} \cdot - 1 + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.13.2

Combine $\frac{x}{3 {(12 - x)}^{\frac{2}{3}}}$ and $- 1$ .

$\frac{x \cdot - 1}{3 {(12 - x)}^{\frac{2}{3}}} + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.13.3

Simplify the expression.

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Step 2.1.13.3.1

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{3 {(12 - x)}^{\frac{2}{3}}} + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.13.3.2

Rewrite $- 1 x$ as $- x$ .

$\frac{- x}{3 {(12 - x)}^{\frac{2}{3}}} + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.13.3.3

Move the negative in front of the fraction.

$- \frac{x}{3 {(12 - x)}^{\frac{2}{3}}} + {(12 - x)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 2.1.14

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

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

Step 2.1.15

Multiply ${(12 - x)}^{\frac{1}{3}}$ by $1$ .

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

Step 2.1.16

To write ${(12 - x)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(12 - x)}^{\frac{2}{3}}}{3 {(12 - x)}^{\frac{2}{3}}}$ .

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

Step 2.1.17

Combine ${(12 - x)}^{\frac{1}{3}}$ and $\frac{3 {(12 - x)}^{\frac{2}{3}}}{3 {(12 - x)}^{\frac{2}{3}}}$ .

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

Step 2.1.18

Combine the numerators over the common denominator.

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

Step 2.1.19

Multiply ${(12 - x)}^{\frac{1}{3}}$ by ${(12 - x)}^{\frac{2}{3}}$ by adding the exponents.

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Step 2.1.19.1

Move ${(12 - x)}^{\frac{2}{3}}$ .

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

Step 2.1.19.2

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

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

Step 2.1.19.3

Combine the numerators over the common denominator.

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

Step 2.1.19.4

Add $2$ and $1$ .

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

Step 2.1.19.5

Divide $3$ by $3$ .

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

Step 2.1.20

Simplify ${(12 - x)}^{1} \cdot 3$ .

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

Step 2.1.21

Move $3$ to the left of $12 - x$ .

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

Step 2.1.22

Simplify.

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Step 2.1.22.1

Apply the distributive property.

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

Step 2.1.22.2

Simplify the numerator.

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Step 2.1.22.2.1

Simplify each term.

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Step 2.1.22.2.1.1

Multiply $3$ by $12$ .

$\frac{- x + 36 + 3 (- x)}{3 {(12 - x)}^{\frac{2}{3}}}$

Step 2.1.22.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- x + 36 - 3 x}{3 {(12 - x)}^{\frac{2}{3}}}$

Step 2.1.22.2.2

Subtract $3 x$ from $- x$ .

$\frac{- 4 x + 36}{3 {(12 - x)}^{\frac{2}{3}}}$

Step 2.1.22.3

Factor $4$ out of $- 4 x + 36$ .

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Step 2.1.22.3.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 36}{3 {(12 - x)}^{\frac{2}{3}}}$

Step 2.1.22.3.2

Factor $4$ out of $36$ .

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

Step 2.1.22.3.3

Factor $4$ out of $4 (- x) + 4 (9)$ .

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

Step 2.1.22.4

Factor $- 1$ out of $- x$ .

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

Step 2.1.22.5

Rewrite $9$ as $- 1 (- 9)$ .

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

Step 2.1.22.6

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

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

Step 2.1.22.7

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

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

Step 2.1.22.8

Move the negative in front of the fraction.

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4 (x - 9)}{3 {(12 - x)}^{\frac{2}{3}}}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [\frac{x - 9}{{(12 - x)}^{\frac{2}{3}}}]$ .

$- \frac{4}{3} \frac{d}{d x} [\frac{x - 9}{{(12 - x)}^{\frac{2}{3}}}]$

Step 2.2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 9$ and $g (x) = {(12 - x)}^{\frac{2}{3}}$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} \frac{d}{d x} [x - 9] - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{({(12 - x)}^{\frac{2}{3}})}^{2}}$

Step 2.2.3

Differentiate.

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Step 2.2.3.1

Multiply the exponents in ${({(12 - x)}^{\frac{2}{3}})}^{2}$ .

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Step 2.2.3.1.1

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

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} \frac{d}{d x} [x - 9] - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{2}{3} \cdot 2}}$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

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

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} \frac{d}{d x} [x - 9] - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{2 \cdot 2}{3}}}$

Step 2.2.3.1.2.2

Multiply $2$ by $2$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} \frac{d}{d x} [x - 9] - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} (1 + 0) - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.3.5

Simplify the expression.

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Step 2.2.3.5.1

Add $1$ and $0$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} \cdot 1 - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.3.5.2

Multiply ${(12 - x)}^{\frac{2}{3}}$ by $1$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) \frac{d}{d x} [{(12 - x)}^{\frac{2}{3}}]}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 12 - x$ .

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Step 2.2.4.1

To apply the Chain Rule, set $u$ as $12 - x$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.4.2

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

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.4.3

Replace all occurrences of $u$ with $12 - x$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{\frac{2}{3} - 1} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.5

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

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.6

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

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

Step 2.2.7

Combine the numerators over the common denominator.

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.8

Simplify the numerator.

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Step 2.2.8.1

Multiply $- 1$ by $3$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{\frac{2 - 3}{3}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.8.2

Subtract $3$ from $2$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{\frac{- 1}{3}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.9

Combine fractions.

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Step 2.2.9.1

Move the negative in front of the fraction.

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3} {(12 - x)}^{- \frac{1}{3}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.9.2

Combine $\frac{2}{3}$ and ${(12 - x)}^{- \frac{1}{3}}$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2 {(12 - x)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.9.3

Move ${(12 - x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3 {(12 - x)}^{\frac{1}{3}}} \frac{d}{d x} [12 - x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.10

By the Sum Rule, the derivative of $12 - x$ with respect to $x$ is $\frac{d}{d x} [12] + \frac{d}{d x} [- x]$ .

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

Step 2.2.11

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3 {(12 - x)}^{\frac{1}{3}}} (0 + \frac{d}{d x} [- x]))}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.12

Add $0$ and $\frac{d}{d x} [- x]$ .

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3 {(12 - x)}^{\frac{1}{3}}} \frac{d}{d x} [- x])}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.13

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

$- \frac{4}{3} \cdot \frac{{(12 - x)}^{\frac{2}{3}} - (x - 9) (\frac{2}{3 {(12 - x)}^{\frac{1}{3}}} (- \frac{d}{d x} [x]))}{{(12 - x)}^{\frac{4}{3}}}$

Step 2.2.14

Multiply.

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Step 2.2.14.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.14.2

Multiply $x - 9$ by $1$ .

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

Step 2.2.15

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

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

Step 2.2.16

Combine fractions.

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Step 2.2.16.1

Multiply $x - 9$ by $1$ .

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

Step 2.2.16.2

Multiply $\frac{{(12 - x)}^{\frac{2}{3}} + (x - 9) \frac{2}{3 {(12 - x)}^{\frac{1}{3}}}}{{(12 - x)}^{\frac{4}{3}}}$ by $\frac{4}{3}$ .

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

Step 2.2.16.3

Reorder.

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Step 2.2.16.3.1

Move $4$ to the left of ${(12 - x)}^{\frac{2}{3}} + (x - 9) \frac{2}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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

Step 2.2.16.3.2

Move $3$ to the left of ${(12 - x)}^{\frac{4}{3}}$ .

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

Step 2.2.17

Simplify.

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Step 2.2.17.1

Apply the distributive property.

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

Step 2.2.17.2

Simplify the numerator.

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Step 2.2.17.2.1

Factor $4$ out of $4 {(12 - x)}^{\frac{2}{3}} + 4 (x - 9) \frac{2}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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Step 2.2.17.2.1.1

Factor $4$ out of $4 {(12 - x)}^{\frac{2}{3}}$ .

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

Step 2.2.17.2.1.2

Factor $4$ out of $4 (x - 9) \frac{2}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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

Step 2.2.17.2.1.3

Factor $4$ out of $4 ({(12 - x)}^{\frac{2}{3}}) + 4 ((x - 9) \frac{2}{3 {(12 - x)}^{\frac{1}{3}}})$ .

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

Step 2.2.17.2.2

Multiply $x - 9$ by $\frac{2}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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

Step 2.2.17.2.3

Move $2$ to the left of $x - 9$ .

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

Step 2.2.17.2.4

To write ${(12 - x)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(12 - x)}^{\frac{1}{3}}}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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

Step 2.2.17.2.5

Combine ${(12 - x)}^{\frac{2}{3}}$ and $\frac{3 {(12 - x)}^{\frac{1}{3}}}{3 {(12 - x)}^{\frac{1}{3}}}$ .

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

Step 2.2.17.2.6

Combine the numerators over the common denominator.

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

Step 2.2.17.2.7

Rewrite $\frac{{(12 - x)}^{\frac{2}{3}} (3 {(12 - x)}^{\frac{1}{3}}) + 2 (x - 9)}{3 {(12 - x)}^{\frac{1}{3}}}$ in a factored form.

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Step 2.2.17.2.7.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.17.2.7.2

Multiply ${(12 - x)}^{\frac{2}{3}}$ by ${(12 - x)}^{\frac{1}{3}}$ by adding the exponents.

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Step 2.2.17.2.7.2.1

Move ${(12 - x)}^{\frac{1}{3}}$ .

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

Step 2.2.17.2.7.2.2

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

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

Step 2.2.17.2.7.2.3

Combine the numerators over the common denominator.

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

Step 2.2.17.2.7.2.4

Add $1$ and $2$ .

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

Step 2.2.17.2.7.2.5

Divide $3$ by $3$ .

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

Step 2.2.17.2.7.3

Simplify $3 {(12 - x)}^{1}$ .

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

Step 2.2.17.2.7.4

Apply the distributive property.

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

Step 2.2.17.2.7.5

Multiply $3$ by $12$ .

$- \frac{4 \frac{36 + 3 (- x) + 2 (x - 9)}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.2.7.6

Multiply $- 1$ by $3$ .

$- \frac{4 \frac{36 - 3 x + 2 (x - 9)}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.2.7.7

Apply the distributive property.

$- \frac{4 \frac{36 - 3 x + 2 x + 2 \cdot - 9}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.2.7.8

Multiply $2$ by $- 9$ .

$- \frac{4 \frac{36 - 3 x + 2 x - 18}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.2.7.9

Subtract $18$ from $36$ .

$- \frac{4 \frac{- 3 x + 2 x + 18}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.2.7.10

Add $- 3 x$ and $2 x$ .

$- \frac{4 \frac{- x + 18}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.3

Combine terms.

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Step 2.2.17.3.1

Combine $4$ and $\frac{- x + 18}{3 {(12 - x)}^{\frac{1}{3}}}$ .

$- \frac{\frac{4 (- x + 18)}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.3.2

Rewrite $\frac{\frac{4 (- x + 18)}{3 {(12 - x)}^{\frac{1}{3}}}}{3 {(12 - x)}^{\frac{4}{3}}}$ as a product.

$- (\frac{4 (- x + 18)}{3 {(12 - x)}^{\frac{1}{3}}} \cdot \frac{1}{3 {(12 - x)}^{\frac{4}{3}}})$

Step 2.2.17.3.3

Multiply $\frac{4 (- x + 18)}{3 {(12 - x)}^{\frac{1}{3}}}$ by $\frac{1}{3 {(12 - x)}^{\frac{4}{3}}}$ .

$- \frac{4 (- x + 18)}{3 {(12 - x)}^{\frac{1}{3}} (3 {(12 - x)}^{\frac{4}{3}})}$

Step 2.2.17.3.4

Multiply $3$ by $3$ .

$- \frac{4 (- x + 18)}{9 {(12 - x)}^{\frac{1}{3}} {(12 - x)}^{\frac{4}{3}}}$

Step 2.2.17.3.5

Multiply ${(12 - x)}^{\frac{1}{3}}$ by ${(12 - x)}^{\frac{4}{3}}$ by adding the exponents.

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Step 2.2.17.3.5.1

Move ${(12 - x)}^{\frac{4}{3}}$ .

$- \frac{4 (- x + 18)}{9 ({(12 - x)}^{\frac{4}{3}} {(12 - x)}^{\frac{1}{3}})}$

Step 2.2.17.3.5.2

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

$- \frac{4 (- x + 18)}{9 {(12 - x)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 2.2.17.3.5.3

Combine the numerators over the common denominator.

$- \frac{4 (- x + 18)}{9 {(12 - x)}^{\frac{4 + 1}{3}}}$

Step 2.2.17.3.5.4

Add $4$ and $1$ .

$- \frac{4 (- x + 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.4

Factor $- 1$ out of $- x$ .

$- \frac{4 (- (x) + 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.5

Rewrite $18$ as $- 1 (- 18)$ .

$- \frac{4 (- (x) - 1 (- 18))}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.6

Factor $- 1$ out of $- (x) - 1 (- 18)$ .

$- \frac{4 (- (x - 18))}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.7

Rewrite $- (x - 18)$ as $- 1 (x - 18)$ .

$- \frac{4 (- 1 (x - 18))}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.8

Move the negative in front of the fraction.

$- - \frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.9

Multiply $- 1$ by $- 1$ .

$1 \frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.2.17.10

Multiply $\frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$ by $1$ .

$f'' (x) = \frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$ .

$\frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{4 (x - 18)}{9 {(12 - x)}^{\frac{5}{3}}} = 0$

Step 3.2

Set the numerator equal to zero.

$4 (x - 18) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Divide each term in $4 (x - 18) = 0$ by $4$ and simplify.

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Step 3.3.1.1

Divide each term in $4 (x - 18) = 0$ by $4$ .

$\frac{4 (x - 18)}{4} = \frac{0}{4}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $4$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x - 18)}{4} = \frac{0}{4}$

Step 3.3.1.2.1.2

Divide $x - 18$ by $1$ .

$x - 18 = \frac{0}{4}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Divide $0$ by $4$ .

$x - 18 = 0$

Step 3.3.2

Add $18$ to both sides of the equation.

$x = 18$

Step 4

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

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Step 4.1

Substitute $18$ in $f (x) = x {(12 - x)}^{\frac{1}{3}}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $18$ in the expression.

$f (18) = (18) {(12 - (18))}^{\frac{1}{3}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 1$ by $18$ .

$f (18) = 18 {(12 - 18)}^{\frac{1}{3}}$

Step 4.1.2.2

Subtract $18$ from $12$ .

$f (18) = 18 {(- 6)}^{\frac{1}{3}}$

Step 4.1.2.3

The final answer is $18 {(- 6)}^{\frac{1}{3}}$ .

$18 {(- 6)}^{\frac{1}{3}}$

Step 4.2

The point found by substituting $18$ in $f (x) = x {(12 - x)}^{\frac{1}{3}}$ is $(18, 18 {(- 6)}^{\frac{1}{3}})$ . This point can be an inflection point.

$(18, 18 {(- 6)}^{\frac{1}{3}})$

Step 5

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

$(- \infty, 18) \cup (18, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 18)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $17.9$ in the expression.

$f'' (17.9) = \frac{4 ((17.9) - 18)}{9 {(12 - (17.9))}^{\frac{5}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Subtract $18$ from $17.9$ .

$f'' (17.9) = \frac{4 \cdot - 0.1}{9 {(12 - (17.9))}^{\frac{5}{3}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $- 1$ by $17.9$ .

$f'' (17.9) = \frac{4 \cdot - 0.1}{9 {(12 - 17.9)}^{\frac{5}{3}}}$

Step 6.2.2.2

Subtract $17.9$ from $12$ .

$f'' (17.9) = \frac{4 \cdot - 0.1}{9 {(- 5.9)}^{\frac{5}{3}}}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $4$ by $- 0.1$ .

$f'' (17.9) = \frac{- 0.4}{9 {(- 5.9)}^{\frac{5}{3}}}$

Step 6.2.3.2

Move the negative in front of the fraction.

$f'' (17.9) = - \frac{0.4}{9 {(- 5.9)}^{\frac{5}{3}}}$

Step 6.2.4

The final answer is $- \frac{0.4}{9 {(- 5.9)}^{\frac{5}{3}}}$ .

$- \frac{0.4}{9 {(- 5.9)}^{\frac{5}{3}}}$

Step 6.3

At $17.9$ , the second derivative is $0.00230708$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 18)$ .

Increasing on $(- \infty, 18)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(18, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $18.1$ in the expression.

$f'' (18.1) = \frac{4 ((18.1) - 18)}{9 {(12 - (18.1))}^{\frac{5}{3}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Subtract $18$ from $18.1$ .

$f'' (18.1) = \frac{4 \cdot 0.1}{9 {(12 - (18.1))}^{\frac{5}{3}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Multiply $- 1$ by $18.1$ .

$f'' (18.1) = \frac{4 \cdot 0.1}{9 {(12 - 18.1)}^{\frac{5}{3}}}$

Step 7.2.2.2

Subtract $18.1$ from $12$ .

$f'' (18.1) = \frac{4 \cdot 0.1}{9 {(- 6.1)}^{\frac{5}{3}}}$

Step 7.2.3

Multiply $4$ by $0.1$ .

$f'' (18.1) = \frac{0.4}{9 {(- 6.1)}^{\frac{5}{3}}}$

Step 7.2.4

The final answer is $\frac{0.4}{9 {(- 6.1)}^{\frac{5}{3}}}$ .

$\frac{0.4}{9 {(- 6.1)}^{\frac{5}{3}}}$

Step 7.3

At $18.1$ , the second derivative is $- 0.0021824$ . Since this is negative, the second derivative is decreasing on the interval $(18, \infty)$

Decreasing on $(18, \infty)$ 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 $(18, 18 {(- 6)}^{\frac{1}{3}})$ .

$(18, 18 {(- 6)}^{\frac{1}{3}})$

Step 9