Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Since $18$ is constant with respect to $x$ , the derivative of $18 (x - 3) {(x - 1)}^{\frac{2}{3}}$ with respect to $x$ is $18 \frac{d}{d x} [(x - 3) {(x - 1)}^{\frac{2}{3}}]$ .

$18 \frac{d}{d x} [(x - 3) {(x - 1)}^{\frac{2}{3}}]$

Step 1.2

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

$18 ((x - 3) \frac{d}{d x} [{(x - 1)}^{\frac{2}{3}}] + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = x - 1$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$18 ((x - 3) (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.3.2

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

$18 ((x - 3) (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.3.3

Replace all occurrences of $u$ with $x - 1$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.4

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

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.5

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

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.6

Combine the numerators over the common denominator.

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $3$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2 - 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.7.2

Subtract $3$ from $2$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{- 1}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{- \frac{1}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.8.2

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

$18 ((x - 3) (\frac{2 {(x - 1)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.8.3

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

$18 ((x - 3) (\frac{2}{3 {(x - 1)}^{\frac{1}{3}}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.9

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.10

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (1 + \frac{d}{d x} [- 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.11

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (1 + 0) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} \cdot 1 + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.12.2

Multiply $x - 3$ by $1$ .

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 1.13

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]))$

Step 1.14

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (1 + \frac{d}{d x} [- 3]))$

Step 1.15

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (1 + 0))$

Step 1.16

Simplify the expression.

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Step 1.16.1

Add $1$ and $0$ .

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

Step 1.16.2

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

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

Step 1.17

Simplify.

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Step 1.17.1

Apply the distributive property.

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

Step 1.17.2

Combine terms.

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Step 1.17.2.1

Combine $\frac{2}{3 {(x - 1)}^{\frac{1}{3}}}$ and $18$ .

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

Step 1.17.2.2

Multiply $2$ by $18$ .

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

Step 1.17.2.3

Factor $3$ out of $36$ .

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

Step 1.17.2.4

Cancel the common factors.

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Step 1.17.2.4.1

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

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

Step 1.17.2.4.2

Cancel the common factor.

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

Step 1.17.2.4.3

Rewrite the expression.

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

Step 1.17.3

Reorder terms.

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

Step 1.17.4

Simplify each term.

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Step 1.17.4.1

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

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

Step 1.17.4.2

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

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

Step 1.17.5

To write $18 {(x - 1)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{1}{3}}}$ .

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

Step 1.17.6

Combine the numerators over the common denominator.

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

Step 1.17.7

Simplify the numerator.

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Step 1.17.7.1

Factor $6$ out of $12 (x - 3) + 18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}$ .

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Step 1.17.7.1.1

Factor $6$ out of $12 (x - 3)$ .

$\frac{6 (2 (x - 3)) + 18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{1}{3}}}$

Step 1.17.7.1.2

Factor $6$ out of $18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}$ .

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

Step 1.17.7.1.3

Factor $6$ out of $6 (2 (x - 3)) + 6 (3 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}})$ .

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

Step 1.17.7.2

Apply the distributive property.

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

Step 1.17.7.3

Multiply $2$ by $- 3$ .

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

Step 1.17.7.4

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

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Step 1.17.7.4.1

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

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

Step 1.17.7.4.2

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

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

Step 1.17.7.4.3

Combine the numerators over the common denominator.

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

Step 1.17.7.4.4

Add $1$ and $2$ .

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

Step 1.17.7.4.5

Divide $3$ by $3$ .

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

Step 1.17.7.5

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

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

Step 1.17.7.6

Apply the distributive property.

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

Step 1.17.7.7

Multiply $3$ by $- 1$ .

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

Step 1.17.7.8

Add $2 x$ and $3 x$ .

$\frac{6 (5 x - 6 - 3)}{{(x - 1)}^{\frac{1}{3}}}$

Step 1.17.7.9

Subtract $3$ from $- 6$ .

$\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{5 x - 9}{{(x - 1)}^{\frac{1}{3}}}]$ .

$f'' (x) = 6 \frac{d}{d x} (\frac{5 x - 9}{{(x - 1)}^{\frac{1}{3}}})$

Step 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 $f (x) = 5 x - 9$ and $g (x) = {(x - 1)}^{\frac{1}{3}}$ .

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} \frac{d}{d x} (5 x - 9) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{({(x - 1)}^{\frac{1}{3}})}^{2}})$

Step 2.3

Differentiate.

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Step 2.3.1

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

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Step 2.3.1.1

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

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} \frac{d}{d x} (5 x - 9) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{1}{3} \cdot 2}})$

Step 2.3.1.2

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

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} \frac{d}{d x} (5 x - 9) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.2

By the Sum Rule, the derivative of $5 x - 9$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 9]$ .

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} (\frac{d}{d x} (5 x) + \frac{d}{d x} (- 9)) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} (5 \frac{d}{d x} (x) + \frac{d}{d x} (- 9)) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.4

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

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} (5 \cdot 1 + \frac{d}{d x} (- 9)) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.5

Multiply $5$ by $1$ .

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} (5 + \frac{d}{d x} (- 9)) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.6

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

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} (5 + 0) - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $5$ and $0$ .

$f'' (x) = 6 (\frac{{(x - 1)}^{\frac{1}{3}} \cdot 5 - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.3.7.2

Move $5$ to the left of ${(x - 1)}^{\frac{1}{3}}$ .

$f'' (x) = 6 (\frac{5 \cdot {(x - 1)}^{\frac{1}{3}} - (5 x - 9) \frac{d}{d x} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.4

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

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{d}{d u} (u^{\frac{1}{3}}) \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.4.2

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.4.3

Replace all occurrences of $u$ with $x - 1$ .

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{1}{3} - 1} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.5

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.6

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.7

Combine the numerators over the common denominator.

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $3$ .

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{1 - 3}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.8.2

Subtract $3$ from $1$ .

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{\frac{- 2}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.9

Combine fractions.

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Step 2.9.1

Move the negative in front of the fraction.

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3} \cdot {(x - 1)}^{- \frac{2}{3}} \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.9.2

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{{(x - 1)}^{- \frac{2}{3}}}{3} \cdot \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.9.3

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \cdot \frac{d}{d x} (x - 1))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.10

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (- 1)))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.11

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \cdot (1 + \frac{d}{d x} (- 1)))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.12

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \cdot (1 + 0))}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.13

Combine fractions.

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Step 2.13.1

Add $1$ and $0$ .

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}} \cdot 1)}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.13.2

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

$f'' (x) = 6 (\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}})$

Step 2.13.3

Combine $6$ and $\frac{5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} - (5 x - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14

Simplify.

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Step 2.14.1

Apply the distributive property.

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} + (- (5 x) + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.2

Apply the distributive property.

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}}) + 6 ((- (5 x) + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3

Simplify the numerator.

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Step 2.14.3.1

Factor $6$ out of $6 \cdot 5 {(x - 1)}^{\frac{1}{3}} + 6 (- 1 \cdot 5 x - - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}}$ .

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Step 2.14.3.1.1

Factor $6$ out of $6 \cdot 5 {(x - 1)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}}) + 6 (- 1 \cdot (5 x) + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.1.2

Factor $6$ out of $6 (- 1 \cdot 5 x - - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}}) + 6 ((- 1 \cdot (5 x) + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.1.3

Factor $6$ out of $6 (5 {(x - 1)}^{\frac{1}{3}}) + 6 ((- 1 \cdot 5 x - - 9) \frac{1}{3 {(x - 1)}^{\frac{2}{3}}})$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} + (- 1 \cdot (5 x) + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.2

Simplify each term.

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Step 2.14.3.2.1

Multiply $- 1$ by $5$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} + (- 5 x + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.2.2

Multiply $- 1$ by $- 9$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} + (- 5 x + 9) (\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}))}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.3

Multiply $- 5 x + 9$ by $\frac{1}{3 {(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} + \frac{- 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.4

To write $5 {(x - 1)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 1)}^{\frac{2}{3}}}{3 {(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{6 (5 {(x - 1)}^{\frac{1}{3}} \cdot \frac{3 {(x - 1)}^{\frac{2}{3}}}{3 {(x - 1)}^{\frac{2}{3}}} + \frac{- 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.5

Combine $5 {(x - 1)}^{\frac{1}{3}}$ and $\frac{3 {(x - 1)}^{\frac{2}{3}}}{3 {(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{6 (\frac{5 {(x - 1)}^{\frac{1}{3}} (3 {(x - 1)}^{\frac{2}{3}})}{3 {(x - 1)}^{\frac{2}{3}}} + \frac{- 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 (\frac{5 {(x - 1)}^{\frac{1}{3}} (3 {(x - 1)}^{\frac{2}{3}}) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.7

Reorder terms.

$f'' (x) = \frac{6 (\frac{3 \cdot (5 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8

Rewrite $\frac{3 \cdot 5 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}} - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}}$ in a factored form.

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Step 2.14.3.8.1

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

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Step 2.14.3.8.1.1

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

$f'' (x) = \frac{6 (\frac{3 \cdot (5 ({(x - 1)}^{\frac{1}{3}} {(x - 1)}^{\frac{2}{3}})) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.1.2

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

$f'' (x) = \frac{6 (\frac{3 \cdot (5 {(x - 1)}^{\frac{1}{3} + \frac{2}{3}}) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.1.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 (\frac{3 \cdot (5 {(x - 1)}^{\frac{1 + 2}{3}}) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.1.4

Add $1$ and $2$ .

$f'' (x) = \frac{6 (\frac{3 \cdot (5 {(x - 1)}^{\frac{3}{3}}) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.1.5

Divide $3$ by $3$ .

$f'' (x) = \frac{6 (\frac{3 \cdot (5 (x - 1)) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.2

Simplify $3 \cdot 5 {(x - 1)}^{1}$ .

$f'' (x) = \frac{6 (\frac{3 \cdot (5 (x - 1)) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.3

Multiply $3$ by $5$ .

$f'' (x) = \frac{6 (\frac{15 (x - 1) - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.4

Apply the distributive property.

$f'' (x) = \frac{6 (\frac{15 x + 15 \cdot - 1 - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.5

Multiply $15$ by $- 1$ .

$f'' (x) = \frac{6 (\frac{15 x - 15 - 5 x + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.6

Subtract $5 x$ from $15 x$ .

$f'' (x) = \frac{6 (\frac{10 x - 15 + 9}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.7

Add $- 15$ and $9$ .

$f'' (x) = \frac{6 (\frac{10 x - 6}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.8

Factor $2$ out of $10 x - 6$ .

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Step 2.14.3.8.8.1

Factor $2$ out of $10 x$ .

$f'' (x) = \frac{6 (\frac{2 (5 x) - 6}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.8.2

Factor $2$ out of $- 6$ .

$f'' (x) = \frac{6 (\frac{2 (5 x) + 2 (- 3)}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.3.8.8.3

Factor $2$ out of $2 (5 x) + 2 (- 3)$ .

$f'' (x) = \frac{6 (\frac{2 (5 x - 3)}{3 {(x - 1)}^{\frac{2}{3}}})}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4

Combine terms.

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Step 2.14.4.1

Combine $6$ and $\frac{2 (5 x - 3)}{3 {(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{\frac{6 (2 (5 x - 3))}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.2

Multiply $2$ by $6$ .

$f'' (x) = \frac{\frac{12 (5 x - 3)}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.3

Factor $3$ out of $12 (5 x - 3)$ .

$f'' (x) = \frac{\frac{3 (4 (5 x - 3))}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.4

Cancel the common factors.

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Step 2.14.4.4.1

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

$f'' (x) = \frac{\frac{3 (4 (5 x - 3))}{3 ({(x - 1)}^{\frac{2}{3}})}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.4.2

Cancel the common factor.

$f'' (x) = \frac{\frac{3 (4 (5 x - 3))}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.4.3

Rewrite the expression.

$f'' (x) = \frac{\frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.5

Rewrite $\frac{\frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$ as a product.

$f'' (x) = \frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3}}} \cdot \frac{1}{{(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.6

Multiply $\frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3}}}$ by $\frac{1}{{(x - 1)}^{\frac{2}{3}}}$ .

$f'' (x) = \frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{2}{3}}}$

Step 2.14.4.7

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

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Step 2.14.4.7.1

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

$f'' (x) = \frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 2.14.4.7.2

Combine the numerators over the common denominator.

$f'' (x) = \frac{4 (5 x - 3)}{{(x - 1)}^{\frac{2 + 2}{3}}}$

Step 2.14.4.7.3

Add $2$ and $2$ .

$f'' (x) = \frac{4 (5 x - 3)}{{(x - 1)}^{\frac{4}{3}}}$

Step 3

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

$\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $18$ is constant with respect to $x$ , the derivative of $18 (x - 3) {(x - 1)}^{\frac{2}{3}}$ with respect to $x$ is $18 \frac{d}{d x} [(x - 3) {(x - 1)}^{\frac{2}{3}}]$ .

$18 \frac{d}{d x} [(x - 3) {(x - 1)}^{\frac{2}{3}}]$

Step 4.1.2

$18 ((x - 3) \frac{d}{d x} [{(x - 1)}^{\frac{2}{3}}] + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = x - 1$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$18 ((x - 3) (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.3.2

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

$18 ((x - 3) (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.3.3

Replace all occurrences of $u$ with $x - 1$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.4

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

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.5

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

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.6

Combine the numerators over the common denominator.

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $3$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{2 - 3}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.7.2

Subtract $3$ from $2$ .

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{\frac{- 1}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

$18 ((x - 3) (\frac{2}{3} {(x - 1)}^{- \frac{1}{3}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.8.2

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

$18 ((x - 3) (\frac{2 {(x - 1)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.8.3

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

$18 ((x - 3) (\frac{2}{3 {(x - 1)}^{\frac{1}{3}}} \frac{d}{d x} [x - 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.9

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.10

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (1 + \frac{d}{d x} [- 1]) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.11

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} (1 + 0) + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.12

Simplify the expression.

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Step 4.1.12.1

Add $1$ and $0$ .

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} \cdot 1 + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.12.2

Multiply $x - 3$ by $1$ .

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} \frac{d}{d x} [x - 3])$

Step 4.1.13

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]))$

Step 4.1.14

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (1 + \frac{d}{d x} [- 3]))$

Step 4.1.15

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

$18 ((x - 3) \frac{2}{3 {(x - 1)}^{\frac{1}{3}}} + {(x - 1)}^{\frac{2}{3}} (1 + 0))$

Step 4.1.16

Simplify the expression.

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Step 4.1.16.1

Add $1$ and $0$ .

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

Step 4.1.16.2

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

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

Step 4.1.17

Simplify.

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Step 4.1.17.1

Apply the distributive property.

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

Step 4.1.17.2

Combine terms.

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Step 4.1.17.2.1

Combine $\frac{2}{3 {(x - 1)}^{\frac{1}{3}}}$ and $18$ .

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

Step 4.1.17.2.2

Multiply $2$ by $18$ .

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

Step 4.1.17.2.3

Factor $3$ out of $36$ .

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

Step 4.1.17.2.4

Cancel the common factors.

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Step 4.1.17.2.4.1

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

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

Step 4.1.17.2.4.2

Cancel the common factor.

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

Step 4.1.17.2.4.3

Rewrite the expression.

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

Step 4.1.17.3

Reorder terms.

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

Step 4.1.17.4

Simplify each term.

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Step 4.1.17.4.1

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

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

Step 4.1.17.4.2

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

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

Step 4.1.17.5

To write $18 {(x - 1)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{1}{3}}}$ .

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

Step 4.1.17.6

Combine the numerators over the common denominator.

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

Step 4.1.17.7

Simplify the numerator.

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Step 4.1.17.7.1

Factor $6$ out of $12 (x - 3) + 18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}$ .

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Step 4.1.17.7.1.1

Factor $6$ out of $12 (x - 3)$ .

$\frac{6 (2 (x - 3)) + 18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{1}{3}}}$

Step 4.1.17.7.1.2

Factor $6$ out of $18 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}}$ .

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

Step 4.1.17.7.1.3

Factor $6$ out of $6 (2 (x - 3)) + 6 (3 {(x - 1)}^{\frac{2}{3}} {(x - 1)}^{\frac{1}{3}})$ .

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

Step 4.1.17.7.2

Apply the distributive property.

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

Step 4.1.17.7.3

Multiply $2$ by $- 3$ .

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

Step 4.1.17.7.4

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

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Step 4.1.17.7.4.1

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

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

Step 4.1.17.7.4.2

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

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

Step 4.1.17.7.4.3

Combine the numerators over the common denominator.

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

Step 4.1.17.7.4.4

Add $1$ and $2$ .

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

Step 4.1.17.7.4.5

Divide $3$ by $3$ .

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

Step 4.1.17.7.5

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

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

Step 4.1.17.7.6

Apply the distributive property.

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

Step 4.1.17.7.7

Multiply $3$ by $- 1$ .

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

Step 4.1.17.7.8

Add $2 x$ and $3 x$ .

$\frac{6 (5 x - 6 - 3)}{{(x - 1)}^{\frac{1}{3}}}$

Step 4.1.17.7.9

Subtract $3$ from $- 6$ .

$f' (x) = \frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$ .

$\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}} = 0$

Step 5.2

Set the numerator equal to zero.

$6 (5 x - 9) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $6 (5 x - 9) = 0$ by $6$ and simplify.

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Step 5.3.1.1

Divide each term in $6 (5 x - 9) = 0$ by $6$ .

$\frac{6 (5 x - 9)}{6} = \frac{0}{6}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $6$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{6 (5 x - 9)}{6} = \frac{0}{6}$

Step 5.3.1.2.1.2

Divide $5 x - 9$ by $1$ .

$5 x - 9 = \frac{0}{6}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $6$ .

$5 x - 9 = 0$

Step 5.3.2

Add $9$ to both sides of the equation.

$5 x = 9$

Step 5.3.3

Divide each term in $5 x = 9$ by $5$ and simplify.

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Step 5.3.3.1

Divide each term in $5 x = 9$ by $5$ .

$\frac{5 x}{5} = \frac{9}{5}$

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $5$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{9}{5}$

Step 5.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{5}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

$\frac{6 (5 x - 9)}{\sqrt[3]{{(x - 1)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{6 (5 x - 9)}{\sqrt[3]{x - 1}}$

Step 6.2

Set the denominator in $\frac{6 (5 x - 9)}{\sqrt[3]{x - 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x - 1} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

${\sqrt[3]{x - 1}}^{3} = 0^{3}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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Step 6.3.2.2.1.1.1

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

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

Step 6.3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

${(x - 1)}^{1} = 0^{3}$

Step 6.3.2.2.1.2

Simplify.

$x - 1 = 0^{3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x - 1 = 0$

Step 6.3.3

Add $1$ to both sides of the equation.

$x = 1$

Step 7

Critical points to evaluate.

$x = \frac{9}{5}, 1$

Step 8

Evaluate the second derivative at $x = \frac{9}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Cancel the common factor of $5$ .

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Step 9.1.1.1

Cancel the common factor.

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

Step 9.1.1.2

Rewrite the expression.

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

Step 9.1.2

Subtract $3$ from $9$ .

$\frac{4 \cdot 6}{{(\frac{9}{5} - 1)}^{\frac{4}{3}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 \cdot 6}{{(\frac{9}{5} - 1 \cdot \frac{5}{5})}^{\frac{4}{3}}}$

Step 9.2.2

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{4 \cdot 6}{{(\frac{9}{5} + \frac{- 1 \cdot 5}{5})}^{\frac{4}{3}}}$

Step 9.2.3

Combine the numerators over the common denominator.

$\frac{4 \cdot 6}{{(\frac{9 - 1 \cdot 5}{5})}^{\frac{4}{3}}}$

Step 9.2.4

Simplify the numerator.

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Step 9.2.4.1

Multiply $- 1$ by $5$ .

$\frac{4 \cdot 6}{{(\frac{9 - 5}{5})}^{\frac{4}{3}}}$

Step 9.2.4.2

Subtract $5$ from $9$ .

$\frac{4 \cdot 6}{{(\frac{4}{5})}^{\frac{4}{3}}}$

Step 9.2.5

Apply the product rule to $\frac{4}{5}$ .

$\frac{4 \cdot 6}{\frac{4^{\frac{4}{3}}}{5^{\frac{4}{3}}}}$

Step 9.3

Multiply $4$ by $6$ .

$\frac{24}{\frac{4^{\frac{4}{3}}}{5^{\frac{4}{3}}}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$24 \frac{5^{\frac{4}{3}}}{4^{\frac{4}{3}}}$

Step 9.5

Combine $24$ and $\frac{5^{\frac{4}{3}}}{4^{\frac{4}{3}}}$ .

$\frac{24 \cdot 5^{\frac{4}{3}}}{4^{\frac{4}{3}}}$

Step 10

$x = \frac{9}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{9}{5}$ is a local minimum

Step 11

Find the y-value when $x = \frac{9}{5}$ .

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Step 11.1

Replace the variable $x$ with $\frac{9}{5}$ in the expression.

$f (\frac{9}{5}) = 18 ((\frac{9}{5}) - 3) {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2

Simplify the result.

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Step 11.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{9}{5}) = 18 (\frac{9}{5} - 3 \cdot \frac{5}{5}) {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.2

Combine $- 3$ and $\frac{5}{5}$ .

$f (\frac{9}{5}) = 18 (\frac{9}{5} + \frac{- 3 \cdot 5}{5}) {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.3

Combine the numerators over the common denominator.

$f (\frac{9}{5}) = 18 (\frac{9 - 3 \cdot 5}{5}) \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.4

Simplify the numerator.

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Step 11.2.4.1

Multiply $- 3$ by $5$ .

$f (\frac{9}{5}) = 18 (\frac{9 - 15}{5}) \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.4.2

Subtract $15$ from $9$ .

$f (\frac{9}{5}) = 18 (\frac{- 6}{5}) \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.5

Move the negative in front of the fraction.

$f (\frac{9}{5}) = 18 (- \frac{6}{5}) \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.6

Multiply $18 (- \frac{6}{5})$ .

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Step 11.2.6.1

Multiply $- 1$ by $18$ .

$f (\frac{9}{5}) = - 18 (\frac{6}{5}) \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.6.2

Combine $- 18$ and $\frac{6}{5}$ .

$f (\frac{9}{5}) = \frac{- 18 \cdot 6}{5} \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.6.3

Multiply $- 18$ by $6$ .

$f (\frac{9}{5}) = \frac{- 108}{5} \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.7

Move the negative in front of the fraction.

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {((\frac{9}{5}) - 1)}^{\frac{2}{3}}$

Step 11.2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {(\frac{9}{5} - 1 \cdot \frac{5}{5})}^{\frac{2}{3}}$

Step 11.2.9

Combine $- 1$ and $\frac{5}{5}$ .

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {(\frac{9}{5} + \frac{- 1 \cdot 5}{5})}^{\frac{2}{3}}$

Step 11.2.10

Combine the numerators over the common denominator.

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {(\frac{9 - 1 \cdot 5}{5})}^{\frac{2}{3}}$

Step 11.2.11

Simplify the numerator.

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Step 11.2.11.1

Multiply $- 1$ by $5$ .

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {(\frac{9 - 5}{5})}^{\frac{2}{3}}$

Step 11.2.11.2

Subtract $5$ from $9$ .

$f (\frac{9}{5}) = - \frac{108}{5} \cdot {(\frac{4}{5})}^{\frac{2}{3}}$

Step 11.2.12

Apply the product rule to $\frac{4}{5}$ .

$f (\frac{9}{5}) = - \frac{108}{5} \cdot \frac{4^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 11.2.13

Multiply $- \frac{108}{5} \cdot \frac{4^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ .

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Step 11.2.13.1

Multiply $\frac{4^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ by $\frac{108}{5}$ .

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{2}{3}} \cdot 5}$

Step 11.2.13.2

Multiply $5^{\frac{2}{3}}$ by $5$ by adding the exponents.

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Step 11.2.13.2.1

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

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Step 11.2.13.2.1.1

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

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{2}{3}} \cdot 5}$

Step 11.2.13.2.1.2

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

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{2}{3} + 1}}$

Step 11.2.13.2.2

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

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{2}{3} + \frac{3}{3}}}$

Step 11.2.13.2.3

Combine the numerators over the common denominator.

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{2 + 3}{3}}}$

Step 11.2.13.2.4

Add $2$ and $3$ .

$f (\frac{9}{5}) = - \frac{4^{\frac{2}{3}} \cdot 108}{5^{\frac{5}{3}}}$

Step 11.2.14

Move $108$ to the left of $4^{\frac{2}{3}}$ .

$f (\frac{9}{5}) = - \frac{108 \cdot 4^{\frac{2}{3}}}{5^{\frac{5}{3}}}$

Step 11.2.15

The final answer is $- \frac{108 \cdot 4^{\frac{2}{3}}}{5^{\frac{5}{3}}}$ .

$y = - \frac{108 \cdot 4^{\frac{2}{3}}}{5^{\frac{5}{3}}}$

Step 12

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

$\frac{4 (5 (1) - 3)}{{((1) - 1)}^{\frac{4}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Subtract $1$ from $1$ .

$\frac{4 (5 (1) - 3)}{0^{\frac{4}{3}}}$

Step 13.1.2

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

$\frac{4 (5 (1) - 3)}{{(0^{3})}^{\frac{4}{3}}}$

Step 13.1.3

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

$\frac{4 (5 (1) - 3)}{0^{3 (\frac{4}{3})}}$

Step 13.2

Cancel the common factor of $3$ .

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Step 13.2.1

Cancel the common factor.

$\frac{4 (5 (1) - 3)}{0^{3 (\frac{4}{3})}}$

Step 13.2.2

Rewrite the expression.

$\frac{4 (5 (1) - 3)}{0^{4}}$

Step 13.3

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

$\frac{4 (5 (1) - 3)}{0}$

Step 13.4

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

Undefined

Step 14

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

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Step 14.1

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

$(- \infty, 1) \cup (1, \frac{9}{5}) \cup (\frac{9}{5}, \infty)$

Step 14.2

Substitute any number, such as $0$ , from the interval $(- \infty, 1)$ in the first derivative $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = \frac{6 (5 (0) - 9)}{{((0) - 1)}^{\frac{1}{3}}}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify the numerator.

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Step 14.2.2.1.1

Multiply $5$ by $0$ .

$f' (0) = \frac{6 (0 - 9)}{{(0 - 1)}^{\frac{1}{3}}}$

Step 14.2.2.1.2

Subtract $9$ from $0$ .

$f' (0) = \frac{6 \cdot - 9}{{(0 - 1)}^{\frac{1}{3}}}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Subtract $1$ from $0$ .

$f' (0) = \frac{6 \cdot - 9}{{(- 1)}^{\frac{1}{3}}}$

Step 14.2.2.2.2

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

$f' (0) = \frac{6 \cdot - 9}{{({(- 1)}^{3})}^{\frac{1}{3}}}$

Step 14.2.2.2.3

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

$f' (0) = \frac{6 \cdot - 9}{{(- 1)}^{3 (\frac{1}{3})}}$

Step 14.2.2.2.4

Cancel the common factor of $3$ .

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Step 14.2.2.2.4.1

Cancel the common factor.

$f' (0) = \frac{6 \cdot - 9}{{(- 1)}^{3 (\frac{1}{3})}}$

Step 14.2.2.2.4.2

Rewrite the expression.

$f' (0) = \frac{6 \cdot - 9}{- 1}$

Step 14.2.2.2.5

Evaluate the exponent.

$f' (0) = \frac{6 \cdot - 9}{- 1}$

Step 14.2.2.3

Simplify the expression.

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Step 14.2.2.3.1

Multiply $6$ by $- 9$ .

$f' (0) = \frac{- 54}{- 1}$

Step 14.2.2.3.2

Divide $- 54$ by $- 1$ .

$f' (0) = 54$

Step 14.2.2.4

The final answer is $54$ .

$54$

Step 14.3

Substitute any number, such as $1.4$ , from the interval $(1, \frac{9}{5})$ in the first derivative $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $1.4$ in the expression.

$f' (1.4) = \frac{6 (5 (1.4) - 9)}{{((1.4) - 1)}^{\frac{1}{3}}}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify the numerator.

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Step 14.3.2.1.1

Multiply $5$ by $1.4$ .

$f' (1.4) = \frac{6 (7 - 9)}{{(1.4 - 1)}^{\frac{1}{3}}}$

Step 14.3.2.1.2

Subtract $9$ from $7$ .

$f' (1.4) = \frac{6 \cdot - 2}{{(1.4 - 1)}^{\frac{1}{3}}}$

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

Subtract $1$ from $1.4$ .

$f' (1.4) = \frac{6 \cdot - 2}{{0.4}^{\frac{1}{3}}}$

Step 14.3.2.2.2

Raise $0.4$ to the power of $\frac{1}{3}$ .

$f' (1.4) = \frac{6 \cdot - 2}{0.73680629}$

Step 14.3.2.3

Simplify the expression.

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Step 14.3.2.3.1

Multiply $6$ by $- 2$ .

$f' (1.4) = \frac{- 12}{0.73680629}$

Step 14.3.2.3.2

Divide $- 12$ by $0.73680629$ .

$f' (1.4) = - 16.28650569$

Step 14.3.2.4

The final answer is $- 16.28650569$ .

$- 16.28650569$

Step 14.4

Substitute any number, such as $4$ , from the interval $(\frac{9}{5}, \infty)$ in the first derivative $\frac{6 (5 x - 9)}{{(x - 1)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.4.1

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

$f' (4) = \frac{6 (5 (4) - 9)}{{((4) - 1)}^{\frac{1}{3}}}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify the numerator.

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Step 14.4.2.1.1

Multiply $5$ by $4$ .

$f' (4) = \frac{6 (20 - 9)}{{(4 - 1)}^{\frac{1}{3}}}$

Step 14.4.2.1.2

Subtract $9$ from $20$ .

$f' (4) = \frac{6 \cdot 11}{{(4 - 1)}^{\frac{1}{3}}}$

Step 14.4.2.2

Simplify the expression.

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Step 14.4.2.2.1

Subtract $1$ from $4$ .

$f' (4) = \frac{6 \cdot 11}{3^{\frac{1}{3}}}$

Step 14.4.2.2.2

Multiply $6$ by $11$ .

$f' (4) = \frac{66}{3^{\frac{1}{3}}}$

Step 14.4.2.3

The final answer is $\frac{66}{3^{\frac{1}{3}}}$ .

$\frac{66}{3^{\frac{1}{3}}}$

Step 14.5

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

$x = 1$ is a local maximum

Step 14.6

Since the first derivative changed signs from negative to positive around $x = \frac{9}{5}$ , then $x = \frac{9}{5}$ is a local minimum.

$x = \frac{9}{5}$ is a local minimum

Step 14.7

These are the local extrema for $f (x) = 18 (x - 3) {(x - 1)}^{\frac{2}{3}}$ .

$x = 1$ is a local maximum

$x = \frac{9}{5}$ is a local minimum

$x = 1$ is a local maximum

$x = \frac{9}{5}$ is a local minimum

Step 15