Calculus Examples

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

Step 1

Write $(- \frac{1}{4}) {(x - 2)}^{(\frac{8}{3})} + 4 {(x - 2)}^{(\frac{2}{3})}$ as a function.

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

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Combine ${(x - 2)}^{\frac{8}{3}}$ and $\frac{1}{4}$ .

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

Step 2.2.2

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{{(x - 2)}^{\frac{8}{3}}}{4}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [{(x - 2)}^{\frac{8}{3}}]$ .

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

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{8}{3}}$ and $g (x) = x - 2$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x - 2$ .

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

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{8}{3}$ .

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

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x - 2$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} - 1} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.7

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

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} - 1 \cdot \frac{3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.8

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

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.9

Combine the numerators over the common denominator.

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8 - 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.10

Simplify the numerator.

Tap for more steps...

Step 2.2.10.1

Multiply $- 1$ by $3$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8 - 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.10.2

Subtract $3$ from $8$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{5}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.11

Add $1$ and $0$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{5}{3}} \cdot 1) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.12

Combine $\frac{8}{3}$ and ${(x - 2)}^{\frac{5}{3}}$ .

$- \frac{1}{4} (\frac{8 {(x - 2)}^{\frac{5}{3}}}{3} \cdot 1) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.13

Multiply $\frac{8 {(x - 2)}^{\frac{5}{3}}}{3}$ by $1$ .

$- \frac{1}{4} \cdot \frac{8 {(x - 2)}^{\frac{5}{3}}}{3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.14

Multiply $\frac{8 {(x - 2)}^{\frac{5}{3}}}{3}$ by $\frac{1}{4}$ .

$- \frac{8 {(x - 2)}^{\frac{5}{3}}}{3 \cdot 4} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.15

Multiply $3$ by $4$ .

$- \frac{8 {(x - 2)}^{\frac{5}{3}}}{12} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.16

Factor $4$ out of $8 {(x - 2)}^{\frac{5}{3}}$ .

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{12} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.17

Cancel the common factors.

Tap for more steps...

Step 2.2.17.1

Factor $4$ out of $12$ .

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{4 (3)} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.17.2

Cancel the common factor.

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{4 \cdot 3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.2.17.3

Rewrite the expression.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 \frac{d}{d x} [{(x - 2)}^{\frac{2}{3}}]$

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $x - 2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{2}{3}}] \frac{d}{d x} [x - 2])$

Step 2.3.2.2

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {u_{2}}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 2])$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $x - 2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 2])$

Step 2.3.3

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]))$

Step 2.3.4

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (1 + \frac{d}{d x} [- 2]))$

Step 2.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (1 + 0))$

Step 2.3.6

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} (1 + 0))$

Step 2.3.7

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0))$

Step 2.3.8

Combine the numerators over the common denominator.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2 - 1 \cdot 3}{3}} (1 + 0))$

Step 2.3.9

Simplify the numerator.

Tap for more steps...

Step 2.3.9.1

Multiply $- 1$ by $3$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2 - 3}{3}} (1 + 0))$

Step 2.3.9.2

Subtract $3$ from $2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{- 1}{3}} (1 + 0))$

Step 2.3.10

Move the negative in front of the fraction.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{- \frac{1}{3}} (1 + 0))$

Step 2.3.11

Add $1$ and $0$ .

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

Step 2.3.12

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

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

Step 2.3.13

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

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

Step 2.3.14

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

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

Step 2.3.15

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

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

Step 2.3.16

Multiply $4$ by $2$ .

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Combine terms.

Tap for more steps...

Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.1.3

Combine the numerators over the common denominator.

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

Step 2.4.1.4

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

Tap for more steps...

Step 2.4.1.4.1

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

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

Step 2.4.1.4.2

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

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

Step 2.4.1.4.3

Combine the numerators over the common denominator.

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

Step 2.4.1.4.4

Add $1$ and $5$ .

$\frac{- 2 {(x - 2)}^{\frac{6}{3}} + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 2.4.1.4.5

Divide $6$ by $3$ .

$\frac{- 2 {(x - 2)}^{2} + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 2.4.2

Simplify the numerator.

Tap for more steps...

Step 2.4.2.1

Factor $2$ out of $- 2 {(x - 2)}^{2} + 8$ .

Tap for more steps...

Step 2.4.2.1.1

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

$\frac{2 (- {(x - 2)}^{2}) + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 2.4.2.1.2

Factor $2$ out of $8$ .

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

Step 2.4.2.1.3

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

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

Step 2.4.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.2.3

Reorder $- {(x - 2)}^{2}$ and $2^{2}$ .

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

Step 2.4.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x - 2$ .

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

Step 2.4.2.5

Simplify.

Tap for more steps...

Step 2.4.2.5.1

Subtract $2$ from $2$ .

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

Step 2.4.2.5.2

Add $x$ and $0$ .

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

Step 2.4.2.5.3

Apply the distributive property.

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

Step 2.4.2.5.4

Multiply $- 1$ by $- 2$ .

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

Step 2.4.2.5.5

Add $2$ and $2$ .

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

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

Step 2.4.3

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

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

Step 2.4.4

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 2.4.5

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 2.4.6

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 2.4.7

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2 x (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}}$ with respect to $x$ is $- \frac{2}{3} \frac{d}{d x} [\frac{x (x - 4)}{{(x - 2)}^{\frac{1}{3}}}]$ .

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

Step 3.2

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 $f (x) = x (x - 4)$ and $g (x) = {(x - 2)}^{\frac{1}{3}}$ .

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.4

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 $g (x) = x - 4$ .

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

Step 3.5

Differentiate.

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 3.5.4

Simplify the expression.

Tap for more steps...

Step 3.5.4.1

Add $1$ and $0$ .

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

Step 3.5.4.2

Multiply $x$ by $1$ .

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

Step 3.5.5

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

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

Step 3.5.6

Simplify by adding terms.

Tap for more steps...

Step 3.5.6.1

Multiply $x - 4$ by $1$ .

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

Step 3.5.6.2

Add $x$ and $x$ .

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

Step 3.6

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 $g (x) = x - 2$ .

Tap for more steps...

Step 3.6.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 3.6.2

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

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

Step 3.6.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 3.10

Simplify the numerator.

Tap for more steps...

Step 3.10.1

Multiply $- 1$ by $3$ .

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

Step 3.10.2

Subtract $3$ from $1$ .

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

Step 3.11

Combine fractions.

Tap for more steps...

Step 3.11.1

Move the negative in front of the fraction.

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

Step 3.11.2

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

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

Step 3.11.3

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

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

Step 3.11.4

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

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.15

Combine fractions.

Tap for more steps...

Step 3.15.1

Add $1$ and $0$ .

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

Step 3.15.2

Multiply $- 1$ by $1$ .

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

Step 3.15.3

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

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

Step 3.15.4

Reorder.

Tap for more steps...

Step 3.15.4.1

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

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

Step 3.15.4.2

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

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

Step 3.16

Simplify.

Tap for more steps...

Step 3.16.1

Apply the distributive property.

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

Step 3.16.2

Simplify the numerator.

Tap for more steps...

Step 3.16.2.1

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

Tap for more steps...

Step 3.16.2.1.1

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

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

Step 3.16.2.1.2

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

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

Step 3.16.2.2

Apply the distributive property.

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

Step 3.16.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.16.2.4

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

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

Step 3.16.2.5

Apply the distributive property.

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

Step 3.16.2.6

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

Tap for more steps...

Step 3.16.2.6.1

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

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

Step 3.16.2.6.2

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

Step 3.16.2.6.3

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

Step 3.16.2.6.4

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

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

Step 3.16.2.6.5

Add $1$ and $1$ .

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

Step 3.16.2.7

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

Tap for more steps...

Step 3.16.2.7.1

Multiply $- 4$ by $- 1$ .

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

Step 3.16.2.7.2

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

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

Step 3.16.2.8

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

Tap for more steps...

Step 3.16.2.8.1

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

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

Step 3.16.2.8.2

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

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

Step 3.16.2.8.3

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

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

Step 3.16.2.8.4

Combine the numerators over the common denominator.

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

Step 3.16.2.9

To write $- 4 {(x - 2)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 2)}^{\frac{2}{3}}}{3 {(x - 2)}^{\frac{2}{3}}}$ .

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

Step 3.16.2.10

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

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

Step 3.16.2.11

Combine the numerators over the common denominator.

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

Step 3.16.2.12

Combine the numerators over the common denominator.

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

Step 3.16.2.13

Simplify each term.

Tap for more steps...

Step 3.16.2.13.1

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

Tap for more steps...

Step 3.16.2.13.1.1

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

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

Step 3.16.2.13.1.2

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

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

Step 3.16.2.13.1.3

Combine the numerators over the common denominator.

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

Step 3.16.2.13.1.4

Add $2$ and $1$ .

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

Step 3.16.2.13.1.5

Divide $3$ by $3$ .

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

Step 3.16.2.13.2

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

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

Step 3.16.2.13.3

Apply the distributive property.

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

Step 3.16.2.13.4

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.16.2.13.4.1

Move $x$ .

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

Step 3.16.2.13.4.2

Multiply $x$ by $x$ .

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

Step 3.16.2.13.5

Multiply $- 2$ by $2$ .

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

Step 3.16.2.13.6

Apply the distributive property.

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

Step 3.16.2.13.7

Multiply $3$ by $2$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 4 x \cdot 3 - x^{2} - 4 {(x - 2)}^{\frac{1}{3}} (3 {(x - 2)}^{\frac{2}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.8

Multiply $3$ by $- 4$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 {(x - 2)}^{\frac{1}{3}} (3 {(x - 2)}^{\frac{2}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.9

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 {(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{2}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.10

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

Tap for more steps...

Step 3.16.2.13.10.1

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

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 ({(x - 2)}^{\frac{2}{3}} {(x - 2)}^{\frac{1}{3}})) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.10.2

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

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 {(x - 2)}^{\frac{2}{3} + \frac{1}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.10.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 {(x - 2)}^{\frac{2 + 1}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.10.4

Add $2$ and $1$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 {(x - 2)}^{\frac{3}{3}}) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.10.5

Divide $3$ by $3$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 (x - 2)) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.11

Simplify $- 4 \cdot 3 {(x - 2)}^{1}$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 4 \cdot (3 (x - 2)) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.12

Multiply $- 4$ by $3$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 12 (x - 2) + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.13

Apply the distributive property.

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 12 x - 12 \cdot - 2 + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.13.14

Multiply $- 12$ by $- 2$ .

$f'' (x) = - \frac{2 (\frac{6 x^{2} - 12 x - x^{2} - 12 x + 24 + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.14

Subtract $x^{2}$ from $6 x^{2}$ .

$f'' (x) = - \frac{2 (\frac{5 x^{2} - 12 x - 12 x + 24 + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.15

Subtract $12 x$ from $- 12 x$ .

$f'' (x) = - \frac{2 (\frac{5 x^{2} - 24 x + 24 + 4 x}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.2.16

Add $- 24 x$ and $4 x$ .

$f'' (x) = - \frac{2 (\frac{5 x^{2} - 20 x + 24}{3 {(x - 2)}^{\frac{2}{3}}})}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.3

Combine terms.

Tap for more steps...

Step 3.16.3.1

Combine $2$ and $\frac{5 x^{2} - 20 x + 24}{3 {(x - 2)}^{\frac{2}{3}}}$ .

$f'' (x) = - \frac{\frac{2 (5 x^{2} - 20 x + 24)}{3 {(x - 2)}^{\frac{2}{3}}}}{3 {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.3.2

Rewrite $\frac{\frac{2 (5 x^{2} - 20 x + 24)}{3 {(x - 2)}^{\frac{2}{3}}}}{3 {(x - 2)}^{\frac{2}{3}}}$ as a product.

$f'' (x) = - (\frac{2 (5 x^{2} - 20 x + 24)}{3 {(x - 2)}^{\frac{2}{3}}} \cdot \frac{1}{3 {(x - 2)}^{\frac{2}{3}}})$

Step 3.16.3.3

Multiply $\frac{2 (5 x^{2} - 20 x + 24)}{3 {(x - 2)}^{\frac{2}{3}}}$ by $\frac{1}{3 {(x - 2)}^{\frac{2}{3}}}$ .

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{3 {(x - 2)}^{\frac{2}{3}} (3 {(x - 2)}^{\frac{2}{3}})}$

Step 3.16.3.4

Multiply $3$ by $3$ .

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{9 {(x - 2)}^{\frac{2}{3}} {(x - 2)}^{\frac{2}{3}}}$

Step 3.16.3.5

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

Tap for more steps...

Step 3.16.3.5.1

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

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{9 ({(x - 2)}^{\frac{2}{3}} {(x - 2)}^{\frac{2}{3}})}$

Step 3.16.3.5.2

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

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{9 {(x - 2)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 3.16.3.5.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{9 {(x - 2)}^{\frac{2 + 2}{3}}}$

Step 3.16.3.5.4

Add $2$ and $2$ .

$f'' (x) = - \frac{2 (5 x^{2} - 20 x + 24)}{9 {(x - 2)}^{\frac{4}{3}}}$

Step 4

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

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

Step 5

Find the first derivative.

Tap for more steps...

Step 5.1

Find the first derivative.

Tap for more steps...

Step 5.1.1

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

Step 5.1.2

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

Tap for more steps...

Step 5.1.2.1

Combine ${(x - 2)}^{\frac{8}{3}}$ and $\frac{1}{4}$ .

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

Step 5.1.2.2

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{{(x - 2)}^{\frac{8}{3}}}{4}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [{(x - 2)}^{\frac{8}{3}}]$ .

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

Step 5.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{8}{3}}$ and $g (x) = x - 2$ .

Tap for more steps...

Step 5.1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x - 2$ .

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

Step 5.1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{8}{3}$ .

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

Step 5.1.2.3.3

Replace all occurrences of $u_{1}$ with $x - 2$ .

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

Step 5.1.2.4

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

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

Step 5.1.2.5

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

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

Step 5.1.2.6

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} - 1} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.7

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

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} - 1 \cdot \frac{3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.8

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

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.9

Combine the numerators over the common denominator.

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8 - 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.10

Simplify the numerator.

Tap for more steps...

Step 5.1.2.10.1

Multiply $- 1$ by $3$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{8 - 3}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.10.2

Subtract $3$ from $8$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{5}{3}} (1 + 0)) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.11

Add $1$ and $0$ .

$- \frac{1}{4} (\frac{8}{3} {(x - 2)}^{\frac{5}{3}} \cdot 1) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.12

Combine $\frac{8}{3}$ and ${(x - 2)}^{\frac{5}{3}}$ .

$- \frac{1}{4} (\frac{8 {(x - 2)}^{\frac{5}{3}}}{3} \cdot 1) + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.13

Multiply $\frac{8 {(x - 2)}^{\frac{5}{3}}}{3}$ by $1$ .

$- \frac{1}{4} \cdot \frac{8 {(x - 2)}^{\frac{5}{3}}}{3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.14

Multiply $\frac{8 {(x - 2)}^{\frac{5}{3}}}{3}$ by $\frac{1}{4}$ .

$- \frac{8 {(x - 2)}^{\frac{5}{3}}}{3 \cdot 4} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.15

Multiply $3$ by $4$ .

$- \frac{8 {(x - 2)}^{\frac{5}{3}}}{12} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.16

Factor $4$ out of $8 {(x - 2)}^{\frac{5}{3}}$ .

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{12} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.17

Cancel the common factors.

Tap for more steps...

Step 5.1.2.17.1

Factor $4$ out of $12$ .

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{4 (3)} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.17.2

Cancel the common factor.

$- \frac{4 (2 {(x - 2)}^{\frac{5}{3}})}{4 \cdot 3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.2.17.3

Rewrite the expression.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + \frac{d}{d x} [4 {(x - 2)}^{\frac{2}{3}}]$

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 \frac{d}{d x} [{(x - 2)}^{\frac{2}{3}}]$

Step 5.1.3.2

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

Tap for more steps...

Step 5.1.3.2.1

To apply the Chain Rule, set $u_{2}$ as $x - 2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{2}{3}}] \frac{d}{d x} [x - 2])$

Step 5.1.3.2.2

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {u_{2}}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 2])$

Step 5.1.3.2.3

Replace all occurrences of $u_{2}$ with $x - 2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} \frac{d}{d x} [x - 2])$

Step 5.1.3.3

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]))$

Step 5.1.3.4

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (1 + \frac{d}{d x} [- 2]))$

Step 5.1.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1} (1 + 0))$

Step 5.1.3.6

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} (1 + 0))$

Step 5.1.3.7

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

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0))$

Step 5.1.3.8

Combine the numerators over the common denominator.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2 - 1 \cdot 3}{3}} (1 + 0))$

Step 5.1.3.9

Simplify the numerator.

Tap for more steps...

Step 5.1.3.9.1

Multiply $- 1$ by $3$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{2 - 3}{3}} (1 + 0))$

Step 5.1.3.9.2

Subtract $3$ from $2$ .

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{\frac{- 1}{3}} (1 + 0))$

Step 5.1.3.10

Move the negative in front of the fraction.

$- \frac{2 {(x - 2)}^{\frac{5}{3}}}{3} + 4 (\frac{2}{3} {(x - 2)}^{- \frac{1}{3}} (1 + 0))$

Step 5.1.3.11

Add $1$ and $0$ .

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

Step 5.1.3.12

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

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

Step 5.1.3.13

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

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

Step 5.1.3.14

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

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

Step 5.1.3.15

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

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

Step 5.1.3.16

Multiply $4$ by $2$ .

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

Step 5.1.4

Simplify.

Tap for more steps...

Step 5.1.4.1

Combine terms.

Tap for more steps...

Step 5.1.4.1.1

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

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

Step 5.1.4.1.2

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

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

Step 5.1.4.1.3

Combine the numerators over the common denominator.

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

Step 5.1.4.1.4

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

Tap for more steps...

Step 5.1.4.1.4.1

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

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

Step 5.1.4.1.4.2

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

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

Step 5.1.4.1.4.3

Combine the numerators over the common denominator.

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

Step 5.1.4.1.4.4

Add $1$ and $5$ .

$\frac{- 2 {(x - 2)}^{\frac{6}{3}} + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 5.1.4.1.4.5

Divide $6$ by $3$ .

$\frac{- 2 {(x - 2)}^{2} + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 5.1.4.2

Simplify the numerator.

Tap for more steps...

Step 5.1.4.2.1

Factor $2$ out of $- 2 {(x - 2)}^{2} + 8$ .

Tap for more steps...

Step 5.1.4.2.1.1

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

$\frac{2 (- {(x - 2)}^{2}) + 8}{3 {(x - 2)}^{\frac{1}{3}}}$

Step 5.1.4.2.1.2

Factor $2$ out of $8$ .

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

Step 5.1.4.2.1.3

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

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

Step 5.1.4.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.1.4.2.3

Reorder $- {(x - 2)}^{2}$ and $2^{2}$ .

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

Step 5.1.4.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x - 2$ .

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

Step 5.1.4.2.5

Simplify.

Tap for more steps...

Step 5.1.4.2.5.1

Subtract $2$ from $2$ .

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

Step 5.1.4.2.5.2

Add $x$ and $0$ .

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

Step 5.1.4.2.5.3

Apply the distributive property.

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

Step 5.1.4.2.5.4

Multiply $- 1$ by $- 2$ .

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

Step 5.1.4.2.5.5

Add $2$ and $2$ .

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

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

Step 5.1.4.3

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

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

Step 5.1.4.4

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 5.1.4.5

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 5.1.4.6

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 5.1.4.7

Move the negative in front of the fraction.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{(2 x) (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}}$ .

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

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{(2 x) (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}} = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$2 x (x - 4) = 0$

Step 6.3

Solve the equation for $x$ .

Tap for more steps...

Step 6.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0$

Step 6.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.3

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.3.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.3.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6.3.4

The final solution is all the values that make $2 x (x - 4) = 0$ true.

$x = 0, 4$

Step 7

Find the values where the derivative is undefined.

Tap for more steps...

Step 7.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

Anything raised to $1$ is the base itself.

$- \frac{(2 x) (x - 4)}{3 \sqrt[3]{x - 2}}$

Step 7.2

Set the denominator in $\frac{(2 x) (x - 4)}{3 \sqrt[3]{x - 2}}$ equal to $0$ to find where the expression is undefined.

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

Step 7.3

Solve for $x$ .

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

Simplify each side of the equation.

Tap for more steps...

Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 2}$ as ${(x - 2)}^{\frac{1}{3}}$ .

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

Step 7.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.2.1

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

Tap for more steps...

Step 7.3.2.2.1.1

Apply the product rule to $3 {(x - 2)}^{\frac{1}{3}}$ .

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

Step 7.3.2.2.1.2

Raise $3$ to the power of $3$ .

$27 {({(x - 2)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 7.3.2.2.1.3

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

Tap for more steps...

Step 7.3.2.2.1.3.1

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

$27 {(x - 2)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(x - 2)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 2)}^{1} = 0^{3}$

Step 7.3.2.2.1.4

Simplify.

$27 (x - 2) = 0^{3}$

Step 7.3.2.2.1.5

Apply the distributive property.

$27 x + 27 \cdot - 2 = 0^{3}$

Step 7.3.2.2.1.6

Multiply $27$ by $- 2$ .

$27 x - 54 = 0^{3}$

Step 7.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.2.3.1

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

$27 x - 54 = 0$

Step 7.3.3

Solve for $x$ .

Tap for more steps...

Step 7.3.3.1

Add $54$ to both sides of the equation.

$27 x = 54$

Step 7.3.3.2

Divide each term in $27 x = 54$ by $27$ and simplify.

Tap for more steps...

Step 7.3.3.2.1

Divide each term in $27 x = 54$ by $27$ .

$\frac{27 x}{27} = \frac{54}{27}$

Step 7.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.3.2.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 7.3.3.2.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{54}{27}$

Step 7.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{54}{27}$

Step 7.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.3.2.3.1

Divide $54$ by $27$ .

$x = 2$

Step 8

Critical points to evaluate.

$x = 0, 4, 2$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (5 {(0)}^{2} - 20 \cdot 0 + 24)}{9 {((0) - 2)}^{\frac{4}{3}}}$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

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

$- \frac{2 (5 \cdot 0 - 20 \cdot 0 + 24)}{9 {(0 - 2)}^{\frac{4}{3}}}$

Step 10.1.2

Multiply $5$ by $0$ .

$- \frac{2 (0 - 20 \cdot 0 + 24)}{9 {(0 - 2)}^{\frac{4}{3}}}$

Step 10.1.3

Multiply $- 20$ by $0$ .

$- \frac{2 (0 + 0 + 24)}{9 {(0 - 2)}^{\frac{4}{3}}}$

Step 10.1.4

Add $0$ and $0$ .

$- \frac{2 (0 + 24)}{9 {(0 - 2)}^{\frac{4}{3}}}$

Step 10.1.5

Add $0$ and $24$ .

$- \frac{2 \cdot 24}{9 {(0 - 2)}^{\frac{4}{3}}}$

Step 10.2

Simplify with factoring out.

Tap for more steps...

Step 10.2.1

Subtract $2$ from $0$ .

$- \frac{2 \cdot 24}{9 {(- 2)}^{\frac{4}{3}}}$

Step 10.2.2

Multiply $2$ by $24$ .

$- \frac{48}{9 {(- 2)}^{\frac{4}{3}}}$

Step 10.2.3

Factor $3$ out of $48$ .

$- \frac{3 (16)}{9 {(- 2)}^{\frac{4}{3}}}$

Step 10.3

Cancel the common factors.

Tap for more steps...

Step 10.3.1

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

$- \frac{3 (16)}{3 (3 {(- 2)}^{\frac{4}{3}})}$

Step 10.3.2

Cancel the common factor.

$- \frac{3 \cdot 16}{3 (3 {(- 2)}^{\frac{4}{3}})}$

Step 10.3.3

Rewrite the expression.

$- \frac{16}{3 {(- 2)}^{\frac{4}{3}}}$

Step 11

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 12

Find the y-value when $x = 0$ .

Tap for more steps...

Step 12.1

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

$f (0) = (- \frac{1}{4}) {((0) - 2)}^{\frac{8}{3}} + 4 {((0) - 2)}^{\frac{2}{3}}$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

Subtract $2$ from $0$ .

$f (0) = - \frac{1}{4} \cdot {(- 2)}^{\frac{8}{3}} + 4 {((0) - 2)}^{\frac{2}{3}}$

Step 12.2.1.2

Combine ${(- 2)}^{\frac{8}{3}}$ and $\frac{1}{4}$ .

$f (0) = - \frac{{(- 2)}^{\frac{8}{3}}}{4} + 4 {((0) - 2)}^{\frac{2}{3}}$

Step 12.2.1.3

Subtract $2$ from $0$ .

$f (0) = - \frac{{(- 2)}^{\frac{8}{3}}}{4} + 4 {(- 2)}^{\frac{2}{3}}$

Step 12.2.2

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

$f (0) = - \frac{{(- 2)}^{\frac{8}{3}}}{4} + 4 {(- 2)}^{\frac{2}{3}} \cdot \frac{4}{4}$

Step 12.2.3

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

$f (0) = - \frac{{(- 2)}^{\frac{8}{3}}}{4} + \frac{4 {(- 2)}^{\frac{2}{3}} \cdot 4}{4}$

Step 12.2.4

Simplify the expression.

Tap for more steps...

Step 12.2.4.1

Combine the numerators over the common denominator.

$f (0) = \frac{- {(- 2)}^{\frac{8}{3}} + 4 {(- 2)}^{\frac{2}{3}} \cdot 4}{4}$

Step 12.2.4.2

Multiply $4$ by $4$ .

$f (0) = \frac{- {(- 2)}^{\frac{8}{3}} + 16 {(- 2)}^{\frac{2}{3}}}{4}$

Step 12.2.5

Factor $- 1$ out of $- {(- 2)}^{\frac{8}{3}}$ .

$f (0) = \frac{- ({(- 2)}^{\frac{8}{3}}) + 16 {(- 2)}^{\frac{2}{3}}}{4}$

Step 12.2.6

Factor $- 1$ out of $16 {(- 2)}^{\frac{2}{3}}$ .

$f (0) = \frac{- ({(- 2)}^{\frac{8}{3}}) - (- 16 {(- 2)}^{\frac{2}{3}})}{4}$

Step 12.2.7

Factor $- 1$ out of $- ({(- 2)}^{\frac{8}{3}}) - (- 16 {(- 2)}^{\frac{2}{3}})$ .

$f (0) = \frac{- ({(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}})}{4}$

Step 12.2.8

Simplify the expression.

Tap for more steps...

Step 12.2.8.1

Rewrite $- ({(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}})$ as $- 1 ({(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}})$ .

$f (0) = \frac{- 1 ({(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}})}{4}$

Step 12.2.8.2

Move the negative in front of the fraction.

$f (0) = - \frac{{(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}}}{4}$

Step 12.2.9

The final answer is $- \frac{{(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}}}{4}$ .

$y = - \frac{{(- 2)}^{\frac{8}{3}} - 16 {(- 2)}^{\frac{2}{3}}}{4}$

Step 13

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (5 {(4)}^{2} - 20 \cdot 4 + 24)}{9 {((4) - 2)}^{\frac{4}{3}}}$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Simplify the numerator.

Tap for more steps...

Step 14.1.1

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

$- \frac{2 (5 \cdot 16 - 20 \cdot 4 + 24)}{9 {(4 - 2)}^{\frac{4}{3}}}$

Step 14.1.2

Multiply $5$ by $16$ .

$- \frac{2 (80 - 20 \cdot 4 + 24)}{9 {(4 - 2)}^{\frac{4}{3}}}$

Step 14.1.3

Multiply $- 20$ by $4$ .

$- \frac{2 (80 - 80 + 24)}{9 {(4 - 2)}^{\frac{4}{3}}}$

Step 14.1.4

Subtract $80$ from $80$ .

$- \frac{2 (0 + 24)}{9 {(4 - 2)}^{\frac{4}{3}}}$

Step 14.1.5

Add $0$ and $24$ .

$- \frac{2 \cdot 24}{9 {(4 - 2)}^{\frac{4}{3}}}$

Step 14.2

Simplify with factoring out.

Tap for more steps...

Step 14.2.1

Subtract $2$ from $4$ .

$- \frac{2 \cdot 24}{9 \cdot 2^{\frac{4}{3}}}$

Step 14.2.2

Multiply $2$ by $24$ .

$- \frac{48}{9 \cdot 2^{\frac{4}{3}}}$

Step 14.2.3

Factor $3$ out of $48$ .

$- \frac{3 (16)}{9 \cdot 2^{\frac{4}{3}}}$

Step 14.3

Cancel the common factors.

Tap for more steps...

Step 14.3.1

Factor $3$ out of $9 \cdot 2^{\frac{4}{3}}$ .

$- \frac{3 (16)}{3 (3 \cdot 2^{\frac{4}{3}})}$

Step 14.3.2

Cancel the common factor.

$- \frac{3 \cdot 16}{3 (3 \cdot 2^{\frac{4}{3}})}$

Step 14.3.3

Rewrite the expression.

$- \frac{16}{3 \cdot 2^{\frac{4}{3}}}$

Step 15

$x = 4$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 4$ is a local maximum

Step 16

Find the y-value when $x = 4$ .

Tap for more steps...

Step 16.1

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

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

Step 16.2

Simplify the result.

Tap for more steps...

Step 16.2.1

Simplify each term.

Tap for more steps...

Step 16.2.1.1

Subtract $2$ from $4$ .

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

Step 16.2.1.2

Combine $2^{\frac{8}{3}}$ and $\frac{1}{4}$ .

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

Step 16.2.1.3

Subtract $2$ from $4$ .

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 4 \cdot 2^{\frac{2}{3}}$

Step 16.2.1.4

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

Tap for more steps...

Step 16.2.1.4.1

Rewrite $4$ as $2^{2}$ .

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{2} \cdot 2^{\frac{2}{3}}$

Step 16.2.1.4.2

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

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

Step 16.2.1.4.3

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

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{2 \cdot \frac{3}{3} + \frac{2}{3}}$

Step 16.2.1.4.4

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

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{\frac{2 \cdot 3}{3} + \frac{2}{3}}$

Step 16.2.1.4.5

Combine the numerators over the common denominator.

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{\frac{2 \cdot 3 + 2}{3}}$

Step 16.2.1.4.6

Simplify the numerator.

Tap for more steps...

Step 16.2.1.4.6.1

Multiply $2$ by $3$ .

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{\frac{6 + 2}{3}}$

Step 16.2.1.4.6.2

Add $6$ and $2$ .

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

Step 16.2.2

To write $2^{\frac{8}{3}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + 2^{\frac{8}{3}} \cdot \frac{4}{4}$

Step 16.2.3

Combine $2^{\frac{8}{3}}$ and $\frac{4}{4}$ .

$f (4) = - \frac{2^{\frac{8}{3}}}{4} + \frac{2^{\frac{8}{3}} \cdot 4}{4}$

Step 16.2.4

Combine the numerators over the common denominator.

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8}{3}} \cdot 4}{4}$

Step 16.2.5

Multiply $2^{\frac{8}{3}} \cdot 4$ .

Tap for more steps...

Step 16.2.5.1

Rewrite $4$ as $2^{2}$ .

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8}{3}} \cdot 2^{2}}{4}$

Step 16.2.5.2

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

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

Step 16.2.5.3

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

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8}{3} + 2 \cdot \frac{3}{3}}}{4}$

Step 16.2.5.4

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

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8}{3} + \frac{2 \cdot 3}{3}}}{4}$

Step 16.2.5.5

Combine the numerators over the common denominator.

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8 + 2 \cdot 3}{3}}}{4}$

Step 16.2.5.6

Simplify the numerator.

Tap for more steps...

Step 16.2.5.6.1

Multiply $2$ by $3$ .

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{8 + 6}{3}}}{4}$

Step 16.2.5.6.2

Add $8$ and $6$ .

$f (4) = \frac{- 2^{\frac{8}{3}} + 2^{\frac{14}{3}}}{4}$

Step 16.2.6

Factor $- 1$ out of $- 2^{\frac{8}{3}}$ .

$f (4) = \frac{- (2^{\frac{8}{3}}) + 2^{\frac{14}{3}}}{4}$

Step 16.2.7

Factor $- 1$ out of $2^{\frac{14}{3}}$ .

$f (4) = \frac{- (2^{\frac{8}{3}}) - 1 (- 2^{\frac{14}{3}})}{4}$

Step 16.2.8

Factor $- 1$ out of $- (2^{\frac{8}{3}}) - 1 (- 2^{\frac{14}{3}})$ .

$f (4) = \frac{- (2^{\frac{8}{3}} - 2^{\frac{14}{3}})}{4}$

Step 16.2.9

Simplify the expression.

Tap for more steps...

Step 16.2.9.1

Rewrite $- (2^{\frac{8}{3}} - 2^{\frac{14}{3}})$ as $- 1 (2^{\frac{8}{3}} - 2^{\frac{14}{3}})$ .

$f (4) = \frac{- 1 (2^{\frac{8}{3}} - 2^{\frac{14}{3}})}{4}$

Step 16.2.9.2

Move the negative in front of the fraction.

$f (4) = - \frac{2^{\frac{8}{3}} - 2^{\frac{14}{3}}}{4}$

Step 16.2.10

The final answer is $- \frac{2^{\frac{8}{3}} - 2^{\frac{14}{3}}}{4}$ .

$y = - \frac{2^{\frac{8}{3}} - 2^{\frac{14}{3}}}{4}$

Step 17

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 {((2) - 2)}^{\frac{4}{3}}}$

Step 18

Evaluate the second derivative.

Tap for more steps...

Step 18.1

Simplify the expression.

Tap for more steps...

Step 18.1.1

Subtract $2$ from $2$ .

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot 0^{\frac{4}{3}}}$

Step 18.1.2

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

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot {(0^{3})}^{\frac{4}{3}}}$

Step 18.1.3

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

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 18.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 18.2.1

Cancel the common factor.

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 18.2.2

Rewrite the expression.

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot 0^{4}}$

Step 18.3

Simplify the expression.

Tap for more steps...

Step 18.3.1

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

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{9 \cdot 0}$

Step 18.3.2

Multiply $9$ by $0$ .

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{0}$

Step 18.3.3

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

Undefined

$- \frac{2 (5 {(2)}^{2} - 20 \cdot 2 + 24)}{0}$

Step 18.4

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

Undefined

Step 19

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

Tap for more steps...

Step 19.1

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

$(- \infty, 0) \cup (0, 2) \cup (2, 4) \cup (4, \infty)$

Step 19.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $- \frac{(2 x) (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 19.2.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 19.2.2

Simplify the result.

Tap for more steps...

Step 19.2.2.1

Simplify the expression.

Tap for more steps...

Step 19.2.2.1.1

Multiply $2$ by $- 2$ .

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

Step 19.2.2.1.2

Subtract $2$ from $- 2$ .

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

Step 19.2.2.1.3

Subtract $4$ from $- 2$ .

$f' (- 2) = - \frac{- 4 \cdot - 6}{3 {(- 4)}^{\frac{1}{3}}}$

Step 19.2.2.1.4

Multiply $- 4$ by $- 6$ .

$f' (- 2) = - \frac{24}{3 {(- 4)}^{\frac{1}{3}}}$

Step 19.2.2.2

Factor $3$ out of $24$ .

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

Step 19.2.2.3

Cancel the common factors.

Tap for more steps...

Step 19.2.2.3.1

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

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

Step 19.2.2.3.2

Cancel the common factor.

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

Step 19.2.2.3.3

Rewrite the expression.

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

Step 19.2.2.4

The final answer is $- \frac{8}{{(- 4)}^{\frac{1}{3}}}$ .

$- \frac{8}{{(- 4)}^{\frac{1}{3}}}$

Step 19.3

Substitute any number, such as $1$ , from the interval $(0, 2)$ in the first derivative $- \frac{(2 x) (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 19.3.1

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

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

Step 19.3.2

Simplify the result.

Tap for more steps...

Step 19.3.2.1

Multiply $2$ by $1$ .

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

Step 19.3.2.2

Simplify the denominator.

Tap for more steps...

Step 19.3.2.2.1

Subtract $2$ from $1$ .

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

Step 19.3.2.2.2

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

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

Step 19.3.2.2.3

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

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

Step 19.3.2.2.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 19.3.2.2.4.1

Cancel the common factor.

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

Step 19.3.2.2.4.2

Rewrite the expression.

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

Step 19.3.2.2.5

Evaluate the exponent.

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

Step 19.3.2.3

Simplify the expression.

Tap for more steps...

Step 19.3.2.3.1

Subtract $4$ from $1$ .

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

Step 19.3.2.3.2

Multiply $3$ by $- 1$ .

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

Step 19.3.2.3.3

Multiply $2$ by $- 3$ .

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

Step 19.3.2.3.4

Divide $- 6$ by $- 3$ .

$f' (1) = - 1 \cdot 2$

Step 19.3.2.3.5

Multiply $- 1$ by $2$ .

$f' (1) = - 2$

Step 19.3.2.4

The final answer is $- 2$ .

$- 2$

Step 19.4

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

Tap for more steps...

Step 19.4.1

Replace the variable $x$ with $3$ in the expression.

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

Step 19.4.2

Simplify the result.

Tap for more steps...

Step 19.4.2.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 19.4.2.1.1

Cancel the common factor.

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

Step 19.4.2.1.2

Rewrite the expression.

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

Step 19.4.2.1.3

Subtract $4$ from $3$ .

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

Step 19.4.2.2

Simplify the denominator.

Tap for more steps...

Step 19.4.2.2.1

Subtract $2$ from $3$ .

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

Step 19.4.2.2.2

One to any power is one.

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

Step 19.4.2.3

Simplify the expression.

Tap for more steps...

Step 19.4.2.3.1

Multiply $2$ by $- 1$ .

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

Step 19.4.2.3.2

Divide $- 2$ by $1$ .

$f' (3) = 2$

Step 19.4.2.3.3

Multiply $- 1$ by $- 2$ .

$f' (3) = 2$

Step 19.4.2.4

The final answer is $2$ .

$2$

Step 19.5

Substitute any number, such as $6$ , from the interval $(4, \infty)$ in the first derivative $- \frac{(2 x) (x - 4)}{3 {(x - 2)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

Tap for more steps...

Step 19.5.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = - \frac{(2 (6)) ((6) - 4)}{3 {((6) - 2)}^{\frac{1}{3}}}$

Step 19.5.2

Simplify the result.

Tap for more steps...

Step 19.5.2.1

Factor $3$ out of $(2 (6)) ((6) - 4)$ .

$f' (6) = - \frac{3 (2 \cdot ((2) ((6) - 4)))}{3 {((6) - 2)}^{\frac{1}{3}}}$

Step 19.5.2.2

Cancel the common factors.

Tap for more steps...

Step 19.5.2.2.1

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

$f' (6) = - \frac{3 (2 \cdot ((2) ((6) - 4)))}{3 ({((6) - 2)}^{\frac{1}{3}})}$

Step 19.5.2.2.2

Cancel the common factor.

$f' (6) = - \frac{3 (2 \cdot ((2) ((6) - 4)))}{3 {((6) - 2)}^{\frac{1}{3}}}$

Step 19.5.2.2.3

Rewrite the expression.

$f' (6) = - \frac{2 \cdot ((2) ((6) - 4))}{{((6) - 2)}^{\frac{1}{3}}}$

Step 19.5.2.3

Multiply $2$ by $2$ .

$f' (6) = - \frac{4 (6 - 4)}{{(6 - 2)}^{\frac{1}{3}}}$

Step 19.5.2.4

Subtract $2$ from $6$ .

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

Step 19.5.2.5

Subtract $4$ from $6$ .

$f' (6) = - \frac{4 \cdot 2}{4^{\frac{1}{3}}}$

Step 19.5.2.6

Multiply $4$ by $2$ .

$f' (6) = - \frac{8}{4^{\frac{1}{3}}}$

Step 19.5.2.7

The final answer is $- \frac{8}{4^{\frac{1}{3}}}$ .

$- \frac{8}{4^{\frac{1}{3}}}$

Step 19.6

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 19.7

Since the first derivative changed signs from negative to positive around $x = 2$ , then $x = 2$ is a local minimum.

$x = 2$ is a local minimum

Step 19.8

Since the first derivative changed signs from positive to negative around $x = 4$ , then $x = 4$ is a local maximum.

$x = 4$ is a local maximum

Step 19.9

These are the local extrema for $f (x) = (- \frac{1}{4}) {(x - 2)}^{\frac{8}{3}} + 4 {(x - 2)}^{\frac{2}{3}}$ .

$x = 0$ is a local maximum

$x = 2$ is a local minimum

$x = 4$ is a local maximum

$x = 0$ is a local maximum

$x = 2$ is a local minimum

$x = 4$ is a local maximum

Step 20