Calculus Examples

$x \sqrt{x - 4}$

Step 1

Write $x \sqrt{x - 4}$ as a function.

$f (x) = x \sqrt{x - 4}$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 2.2

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

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 4$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $x - 4$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $x - 4$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

Tap for more steps...

Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $1$ .

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

Step 2.8

Combine fractions.

Tap for more steps...

Step 2.8.1

Move the negative in front of the fraction.

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

Step 2.8.2

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

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

Step 2.8.3

Move ${(x - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8.4

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Simplify the expression.

Tap for more steps...

Step 2.12.1

Add $1$ and $0$ .

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

Step 2.12.2

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Combine the numerators over the common denominator.

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

Step 2.18

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

Tap for more steps...

Step 2.18.1

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

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

Step 2.18.2

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

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

Step 2.18.3

Combine the numerators over the common denominator.

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

Step 2.18.4

Add $1$ and $1$ .

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

Step 2.18.5

Divide $2$ by $2$ .

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

Step 2.19

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

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

Step 2.20

Move $2$ to the left of $x - 4$ .

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

Step 2.21

Simplify.

Tap for more steps...

Step 2.21.1

Apply the distributive property.

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

Step 2.21.2

Simplify the numerator.

Tap for more steps...

Step 2.21.2.1

Multiply $2$ by $- 4$ .

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

Step 2.21.2.2

Add $x$ and $2 x$ .

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

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

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

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

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

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

Step 3.3

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

Simplify.

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

Step 3.5

Differentiate.

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Multiply $3$ by $1$ .

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

Step 3.5.5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

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

Step 3.5.6

Simplify the expression.

Tap for more steps...

Step 3.5.6.1

Add $3$ and $0$ .

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

Step 3.5.6.2

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

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

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

Tap for more steps...

Step 3.6.1

To apply the Chain Rule, set $u_{1}$ as $x - 4$ .

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

Step 3.6.2

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

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

Step 3.6.3

Replace all occurrences of $u_{1}$ with $x - 4$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 3.10

Simplify the numerator.

Tap for more steps...

Step 3.10.1

Multiply $- 1$ by $2$ .

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

Step 3.10.2

Subtract $2$ from $1$ .

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

Step 3.11

Combine fractions.

Tap for more steps...

Step 3.11.1

Move the negative in front of the fraction.

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

Step 3.11.2

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

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

Step 3.11.3

Move ${(x - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Combine fractions.

Tap for more steps...

Step 3.15.1

Add $1$ and $0$ .

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

Step 3.15.2

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

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

Step 3.15.3

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

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

Step 3.16

Simplify.

Tap for more steps...

Step 3.16.1

Apply the distributive property.

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

Step 3.16.2

Apply the distributive property.

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

Step 3.16.3

Simplify the numerator.

Tap for more steps...

Step 3.16.3.1

Add parentheses.

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

Step 3.16.3.2

Let $u_{2} = {(x - 4)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x - 4)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.16.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.16.3.2.2

Multiply $u_{2}$ by $u_{2}$ by adding the exponents.

Tap for more steps...

Step 3.16.3.2.2.1

Move $u_{2}$ .

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

Step 3.16.3.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

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

Step 3.16.3.2.3

Multiply $3$ by $2$ .

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

Step 3.16.3.3

Replace all occurrences of $u_{2}$ with ${(x - 4)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{6 {({(x - 4)}^{\frac{1}{2}})}^{2} - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4

Simplify.

Tap for more steps...

Step 3.16.3.4.1

Simplify each term.

Tap for more steps...

Step 3.16.3.4.1.1

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.16.3.4.1.1.1

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

$f'' (x) = \frac{\frac{6 {(x - 4)}^{\frac{1}{2} \cdot 2} - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.16.3.4.1.1.2.1

Cancel the common factor.

$f'' (x) = \frac{\frac{6 {(x - 4)}^{\frac{1}{2} \cdot 2} - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.1.1.2.2

Rewrite the expression.

$f'' (x) = \frac{\frac{6 (x - 4) - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.1.2

Simplify.

$f'' (x) = \frac{\frac{6 (x - 4) - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.1.3

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x + 6 \cdot - 4 - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.1.4

Multiply $6$ by $- 4$ .

$f'' (x) = \frac{\frac{6 x - 24 - 3 x + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.2

Subtract $3 x$ from $6 x$ .

$f'' (x) = \frac{\frac{3 x - 24 + 8}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.3.4.3

Add $- 24$ and $8$ .

$f'' (x) = \frac{\frac{3 x - 16}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x + 2 \cdot - 4}$

Step 3.16.4

Combine terms.

Tap for more steps...

Step 3.16.4.1

Multiply $2$ by $- 4$ .

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

Step 3.16.4.2

Rewrite $\frac{\frac{3 x - 16}{2 {(x - 4)}^{\frac{1}{2}}}}{2 x - 8}$ as a product.

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

Step 3.16.4.3

Multiply $\frac{3 x - 16}{2 {(x - 4)}^{\frac{1}{2}}}$ by $\frac{1}{2 x - 8}$ .

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

Step 3.16.5

Simplify the denominator.

Tap for more steps...

Step 3.16.5.1

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

Tap for more steps...

Step 3.16.5.1.1

Factor $2$ out of $2 x$ .

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

Step 3.16.5.1.2

Factor $2$ out of $- 8$ .

$f'' (x) = \frac{3 x - 16}{2 {(x - 4)}^{\frac{1}{2}} (2 x + 2 \cdot - 4)}$

Step 3.16.5.1.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

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

$f'' (x) = \frac{3 x - 16}{2 {(x - 4)}^{\frac{1}{2}} \cdot (2 (x - 4))}$

Step 3.16.5.2

Combine exponents.

Tap for more steps...

Step 3.16.5.2.1

Multiply $2$ by $2$ .

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

Step 3.16.5.2.2

Raise $x - 4$ to the power of $1$ .

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

Step 3.16.5.2.3

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

$f'' (x) = \frac{3 x - 16}{4 {(x - 4)}^{1 + \frac{1}{2}}}$

Step 3.16.5.2.4

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

$f'' (x) = \frac{3 x - 16}{4 {(x - 4)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 3.16.5.2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{3 x - 16}{4 {(x - 4)}^{\frac{2 + 1}{2}}}$

Step 3.16.5.2.6

Add $2$ and $1$ .

$f'' (x) = \frac{3 x - 16}{4 {(x - 4)}^{\frac{3}{2}}}$

Step 4

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 5.1.2

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

Step 5.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 4$ .

Tap for more steps...

Step 5.1.3.1

To apply the Chain Rule, set $u$ as $x - 4$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Replace all occurrences of $u$ with $x - 4$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

Combine the numerators over the common denominator.

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

Step 5.1.7

Simplify the numerator.

Tap for more steps...

Step 5.1.7.1

Multiply $- 1$ by $2$ .

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

Step 5.1.7.2

Subtract $2$ from $1$ .

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

Step 5.1.8

Combine fractions.

Tap for more steps...

Step 5.1.8.1

Move the negative in front of the fraction.

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

Step 5.1.8.2

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

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

Step 5.1.8.3

Move ${(x - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.1.8.4

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

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

Step 5.1.9

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

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

Step 5.1.10

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

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

Step 5.1.11

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

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

Step 5.1.12

Simplify the expression.

Tap for more steps...

Step 5.1.12.1

Add $1$ and $0$ .

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

Step 5.1.12.2

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

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

Step 5.1.13

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

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

Step 5.1.14

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

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

Step 5.1.15

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

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

Step 5.1.16

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

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

Step 5.1.17

Combine the numerators over the common denominator.

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

Step 5.1.18

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

Tap for more steps...

Step 5.1.18.1

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

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

Step 5.1.18.2

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

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

Step 5.1.18.3

Combine the numerators over the common denominator.

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

Step 5.1.18.4

Add $1$ and $1$ .

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

Step 5.1.18.5

Divide $2$ by $2$ .

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

Step 5.1.19

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

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

Step 5.1.20

Move $2$ to the left of $x - 4$ .

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

Step 5.1.21

Simplify.

Tap for more steps...

Step 5.1.21.1

Apply the distributive property.

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

Step 5.1.21.2

Simplify the numerator.

Tap for more steps...

Step 5.1.21.2.1

Multiply $2$ by $- 4$ .

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

Step 5.1.21.2.2

Add $x$ and $2 x$ .

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 x - 8}{2 {(x - 4)}^{\frac{1}{2}}}$ .

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

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{3 x - 8}{2 {(x - 4)}^{\frac{1}{2}}} = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$3 x - 8 = 0$

Step 6.3

Solve the equation for $x$ .

Tap for more steps...

Step 6.3.1

Add $8$ to both sides of the equation.

$3 x = 8$

Step 6.3.2

Divide each term in $3 x = 8$ by $3$ and simplify.

Tap for more steps...

Step 6.3.2.1

Divide each term in $3 x = 8$ by $3$ .

$\frac{3 x}{3} = \frac{8}{3}$

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{8}{3}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{3}$

Step 6.4

Exclude the solutions that do not make $\frac{3 x - 8}{2 {(x - 4)}^{\frac{1}{2}}} = 0$ true.

$No solution$

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.

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

Step 7.1.2

Anything raised to $1$ is the base itself.

$\frac{3 x - 8}{2 \sqrt{x - 4}}$

Step 7.2

Set the denominator in $\frac{3 x - 8}{2 \sqrt{x - 4}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x - 4} = 0$

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, square both sides of the equation.

${(2 \sqrt{x - 4})}^{2} = 0^{2}$

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 $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 7.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.2.1

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

Tap for more steps...

Step 7.3.2.2.1.1

Apply the product rule to $2 {(x - 4)}^{\frac{1}{2}}$ .

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

Step 7.3.2.2.1.2

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

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

Step 7.3.2.2.1.3

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 7.3.2.2.1.3.1

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

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

Step 7.3.2.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.4

Simplify.

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

Step 7.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 4 = 0^{2}$

Step 7.3.2.2.1.6

Multiply $4$ by $- 4$ .

$4 x - 16 = 0^{2}$

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 $0$ .

$4 x - 16 = 0$

Step 7.3.3

Solve for $x$ .

Tap for more steps...

Step 7.3.3.1

Add $16$ to both sides of the equation.

$4 x = 16$

Step 7.3.3.2

Divide each term in $4 x = 16$ by $4$ and simplify.

Tap for more steps...

Step 7.3.3.2.1

Divide each term in $4 x = 16$ by $4$ .

$\frac{4 x}{4} = \frac{16}{4}$

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 $4$ .

Tap for more steps...

Step 7.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{16}{4}$

Step 7.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{4}$

Step 7.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.3.2.3.1

Divide $16$ by $4$ .

$x = 4$

Step 7.4

Set the radicand in $\sqrt{x - 4}$ less than $0$ to find where the expression is undefined.

$x - 4 < 0$

Step 7.5

Add $4$ to both sides of the inequality.

$x < 4$

Step 7.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 4$

$(- \infty, 4]$

$x \leq 4$

$(- \infty, 4]$

Step 8

Critical points to evaluate.

$x = 4$

Step 9

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.

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

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify the expression.

Tap for more steps...

Step 10.1.1

Subtract $4$ from $4$ .

$\frac{3 (4) - 16}{4 \cdot 0^{\frac{3}{2}}}$

Step 10.1.2

Rewrite $0$ as $0^{2}$ .

$\frac{3 (4) - 16}{4 \cdot {(0^{2})}^{\frac{3}{2}}}$

Step 10.1.3

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

$\frac{3 (4) - 16}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 10.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.1

Cancel the common factor.

$\frac{3 (4) - 16}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 10.2.2

Rewrite the expression.

$\frac{3 (4) - 16}{4 \cdot 0^{3}}$

Step 10.3

Simplify the expression.

Tap for more steps...

Step 10.3.1

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

$\frac{3 (4) - 16}{4 \cdot 0}$

Step 10.3.2

Multiply $4$ by $0$ .

$\frac{3 (4) - 16}{0}$

Step 10.3.3

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

Undefined

$\frac{3 (4) - 16}{0}$

Step 10.4

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

Undefined

Step 11

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 12