Calculus Examples

$y = x \sqrt{4 - x}$

Step 1

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

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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 $g (x) = 4 - x$ .

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

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

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

Step 2.1.1.3.3

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

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

Step 2.1.1.4

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

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

Step 2.1.1.5

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

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

Step 2.1.1.6

Combine the numerators over the common denominator.

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

Step 2.1.1.7

Simplify the numerator.

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Step 2.1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 2.1.1.7.2

Subtract $2$ from $1$ .

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

Step 2.1.1.8

Combine fractions.

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Step 2.1.1.8.1

Move the negative in front of the fraction.

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

Step 2.1.1.8.2

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

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

Step 2.1.1.8.3

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

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

Step 2.1.1.8.4

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

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

Step 2.1.1.9

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

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

Step 2.1.1.10

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

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

Step 2.1.1.11

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

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

Step 2.1.1.12

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

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

Step 2.1.1.13

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

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

Step 2.1.1.14

Combine fractions.

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Step 2.1.1.14.1

Multiply $- 1$ by $1$ .

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

Step 2.1.1.14.2

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

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

Step 2.1.1.14.3

Simplify the expression.

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Step 2.1.1.14.3.1

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

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

Step 2.1.1.14.3.2

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

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

Step 2.1.1.14.3.3

Move the negative in front of the fraction.

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

Step 2.1.1.15

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

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

Step 2.1.1.16

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

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

Step 2.1.1.17

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

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

Step 2.1.1.18

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

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

Step 2.1.1.19

Combine the numerators over the common denominator.

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

Step 2.1.1.20

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

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Step 2.1.1.20.1

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

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

Step 2.1.1.20.2

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

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

Step 2.1.1.20.3

Combine the numerators over the common denominator.

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

Step 2.1.1.20.4

Add $1$ and $1$ .

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

Step 2.1.1.20.5

Divide $2$ by $2$ .

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

Step 2.1.1.21

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

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

Step 2.1.1.22

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

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

Step 2.1.1.23

Simplify.

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Step 2.1.1.23.1

Apply the distributive property.

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

Step 2.1.1.23.2

Simplify the numerator.

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Step 2.1.1.23.2.1

Simplify each term.

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Step 2.1.1.23.2.1.1

Multiply $2$ by $4$ .

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

Step 2.1.1.23.2.1.2

Multiply $- 1$ by $2$ .

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

Step 2.1.1.23.2.2

Subtract $2 x$ from $- x$ .

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

Step 2.1.1.23.3

Factor $- 1$ out of $- 3 x$ .

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

Step 2.1.1.23.4

Rewrite $8$ as $- 1 (- 8)$ .

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

Step 2.1.1.23.5

Factor $- 1$ out of $- (3 x) - 1 (- 8)$ .

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

Step 2.1.1.23.6

Rewrite $- (3 x - 8)$ as $- 1 (3 x - 8)$ .

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

Step 2.1.1.23.7

Move the negative in front of the fraction.

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 3 x - 8$ and $g (x) = {(4 - x)}^{\frac{1}{2}}$ .

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

Cancel the common factor of $2$ .

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Step 2.1.2.3.2.1

Cancel the common factor.

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

Step 2.1.2.3.2.2

Rewrite the expression.

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

Step 2.1.2.4

Simplify.

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

Step 2.1.2.5

Differentiate.

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

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

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

Step 2.1.2.5.3

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

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

Step 2.1.2.5.4

Multiply $3$ by $1$ .

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

Step 2.1.2.5.5

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

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

Step 2.1.2.5.6

Simplify the expression.

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Step 2.1.2.5.6.1

Add $3$ and $0$ .

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

Step 2.1.2.5.6.2

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

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

Step 2.1.2.6

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 $g (x) = 4 - x$ .

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Step 2.1.2.6.1

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

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

Step 2.1.2.6.2

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

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

Step 2.1.2.6.3

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

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

Step 2.1.2.7

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

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

Step 2.1.2.8

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

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

Step 2.1.2.9

Combine the numerators over the common denominator.

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

Step 2.1.2.10

Simplify the numerator.

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Step 2.1.2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.1.2.10.2

Subtract $2$ from $1$ .

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

Step 2.1.2.11

Combine fractions.

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Step 2.1.2.11.1

Move the negative in front of the fraction.

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

Step 2.1.2.11.2

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

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

Step 2.1.2.11.3

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

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

Step 2.1.2.12

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

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

Step 2.1.2.13

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

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

Step 2.1.2.14

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

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

Step 2.1.2.15

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

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

Step 2.1.2.16

Multiply.

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Step 2.1.2.16.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.16.2

Multiply $3 x - 8$ by $1$ .

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

Step 2.1.2.17

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

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

Step 2.1.2.18

Combine fractions.

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Step 2.1.2.18.1

Multiply $3 x - 8$ by $1$ .

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

Step 2.1.2.18.2

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

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

Step 2.1.2.18.3

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

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

Step 2.1.2.19

Simplify.

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Step 2.1.2.19.1

Apply the distributive property.

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

Step 2.1.2.19.2

Simplify the numerator.

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Step 2.1.2.19.2.1

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

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Step 2.1.2.19.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.19.2.1.2

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

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Step 2.1.2.19.2.1.2.1

Move $u_{2}$ .

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

Step 2.1.2.19.2.1.2.2

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

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

Step 2.1.2.19.2.1.3

Multiply $3$ by $2$ .

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

Step 2.1.2.19.2.2

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

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

Step 2.1.2.19.2.3

Simplify.

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Step 2.1.2.19.2.3.1

Simplify each term.

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Step 2.1.2.19.2.3.1.1

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

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Step 2.1.2.19.2.3.1.1.1

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

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

Step 2.1.2.19.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.19.2.3.1.1.2.1

Cancel the common factor.

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

Step 2.1.2.19.2.3.1.1.2.2

Rewrite the expression.

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

Step 2.1.2.19.2.3.1.2

Simplify.

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

Step 2.1.2.19.2.3.1.3

Apply the distributive property.

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

Step 2.1.2.19.2.3.1.4

Multiply $6$ by $4$ .

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

Step 2.1.2.19.2.3.1.5

Multiply $- 1$ by $6$ .

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

Step 2.1.2.19.2.3.2

Subtract $8$ from $24$ .

$f'' (x) = - \frac{\frac{- 6 x + 3 x + 16}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 2.1.2.19.2.3.3

Add $- 6 x$ and $3 x$ .

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

Step 2.1.2.19.3

Combine terms.

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Step 2.1.2.19.3.1

Multiply $2$ by $4$ .

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

Step 2.1.2.19.3.2

Multiply $- 1$ by $2$ .

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

Step 2.1.2.19.3.3

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

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

Step 2.1.2.19.3.4

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

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

Step 2.1.2.19.4

Simplify the denominator.

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Step 2.1.2.19.4.1

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

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Step 2.1.2.19.4.1.1

Factor $2$ out of $8$ .

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

Step 2.1.2.19.4.1.2

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

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

Step 2.1.2.19.4.1.3

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

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

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

Step 2.1.2.19.4.2

Combine exponents.

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Step 2.1.2.19.4.2.1

Multiply $2$ by $2$ .

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

Step 2.1.2.19.4.2.2

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

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

Step 2.1.2.19.4.2.3

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

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

Step 2.1.2.19.4.2.4

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

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

Step 2.1.2.19.4.2.5

Combine the numerators over the common denominator.

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

Step 2.1.2.19.4.2.6

Add $2$ and $1$ .

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

Step 2.1.2.19.5

Factor $- 1$ out of $- 3 x$ .

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

Step 2.1.2.19.6

Rewrite $16$ as $- 1 (- 16)$ .

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

Step 2.1.2.19.7

Factor $- 1$ out of $- (3 x) - 1 (- 16)$ .

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

Step 2.1.2.19.8

Rewrite $- (3 x - 16)$ as $- 1 (3 x - 16)$ .

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

Step 2.1.2.19.9

Move the negative in front of the fraction.

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

Step 2.1.2.19.10

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.19.11

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

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

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{3 x - 16}{4 {(4 - x)}^{\frac{3}{2}}}$ .

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

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{3 x - 16}{4 {(4 - x)}^{\frac{3}{2}}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Set the numerator equal to zero.

$3 x - 16 = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

Add $16$ to both sides of the equation.

$3 x = 16$

Step 2.2.3.2

Divide each term in $3 x = 16$ by $3$ and simplify.

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Step 2.2.3.2.1

Divide each term in $3 x = 16$ by $3$ .

$\frac{3 x}{3} = \frac{16}{3}$

Step 2.2.3.2.2

Simplify the left side.

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Step 2.2.3.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{16}{3}$

Step 2.2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{3}$

Step 2.2.4

Exclude the solutions that do not make $\frac{3 x - 16}{4 {(4 - x)}^{\frac{3}{2}}} = 0$ true.

$No solution$

Step 3

Find the domain of $f (x) = x \sqrt{4 - x}$ .

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Step 3.1

Set the radicand in $\sqrt{4 - x}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x \geq 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $4$ from both sides of the inequality.

$- x \geq - 4$

Step 3.2.2

Divide each term in $- x \geq - 4$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- x \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 4}{- 1}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 4}{- 1}$

Step 3.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 4}{- 1}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 4$ by $- 1$ .

$x \leq 4$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 4]$

Set-Builder Notation:

${x | x \leq 4}$

Interval Notation:

$(- \infty, 4]$

Set-Builder Notation:

${x | x \leq 4}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 4]$

Step 5

Substitute any number from the interval $(- \infty, 4]$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Factor $4$ out of $3 (0)$ .

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

Step 5.2.2

Factor $4$ out of $- 16$ .

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

Step 5.2.3

Factor $4$ out of $4 (3 \cdot (0)) + 4 (- 4)$ .

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

Step 5.2.4

Cancel the common factors.

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Step 5.2.4.1

Factor $4$ out of $4 {(4 - (0))}^{\frac{3}{2}}$ .

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

Step 5.2.4.2

Cancel the common factor.

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

Step 5.2.4.3

Rewrite the expression.

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

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $3$ by $0$ .

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

Step 5.2.5.2

Subtract $4$ from $0$ .

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

Step 5.2.6

Simplify the denominator.

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Step 5.2.6.1

Subtract $0$ from $4$ .

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

Step 5.2.6.2

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

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

Step 5.2.6.3

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

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

Step 5.2.6.4

Cancel the common factor of $2$ .

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Step 5.2.6.4.1

Cancel the common factor.

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

Step 5.2.6.4.2

Rewrite the expression.

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

Step 5.2.6.5

Raise $2$ to the power of $3$ .

$f'' (0) = \frac{- 4}{8}$

Step 5.2.7

Reduce the expression by cancelling the common factors.

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Step 5.2.7.1

Cancel the common factor of $- 4$ and $8$ .

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Step 5.2.7.1.1

Factor $4$ out of $- 4$ .

$f'' (0) = \frac{4 (- 1)}{8}$

Step 5.2.7.1.2

Cancel the common factors.

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Step 5.2.7.1.2.1

Factor $4$ out of $8$ .

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

Step 5.2.7.1.2.2

Cancel the common factor.

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

Step 5.2.7.1.2.3

Rewrite the expression.

$f'' (0) = \frac{- 1}{2}$

Step 5.2.7.2

Move the negative in front of the fraction.

$f'' (0) = - \frac{1}{2}$

Step 5.2.8

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 5.3

The graph is concave down on the interval $(- \infty, 4]$ because $f'' (0)$ is negative.

Concave down on $(- \infty, 4]$ since $f'' (x)$ is negative

Step 6