Calculus Examples

$x \sqrt{9 - x}$

Step 1

Write $x \sqrt{9 - x}$ as a function.

$f (x) = x \sqrt{9 - x}$

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 $\sqrt{9 - x}$ as ${(9 - x)}^{\frac{1}{2}}$ .

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

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 $g (x) = {(9 - x)}^{\frac{1}{2}}$ .

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

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 $g (x) = 9 - x$ .

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Step 2.1.1.3.1

To apply the Chain Rule, set $u$ as $9 - x$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [9 - x]) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

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 $n = \frac{1}{2}$ .

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x]) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 2.1.1.3.3

Replace all occurrences of $u$ with $9 - x$ .

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

Step 2.1.1.4

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

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

Step 2.1.1.5

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

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

Step 2.1.1.6

Combine the numerators over the common denominator.

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

Step 2.1.1.7

Simplify the numerator.

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Step 2.1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 2.1.1.7.2

Subtract $2$ from $1$ .

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

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.

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

Step 2.1.1.8.2

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

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

Step 2.1.1.8.3

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.1.8.4

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

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

Step 2.1.1.9

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

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

Step 2.1.1.10

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

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

Step 2.1.1.11

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

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

Step 2.1.1.12

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

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

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 $n = 1$ .

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

Step 2.1.1.14

Combine fractions.

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Step 2.1.1.14.1

Multiply $- 1$ by $1$ .

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

Step 2.1.1.14.2

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

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

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 $x$ .

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

Step 2.1.1.14.3.2

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

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

Step 2.1.1.14.3.3

Move the negative in front of the fraction.

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

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 $n = 1$ .

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

Step 2.1.1.16

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

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

Step 2.1.1.17

To write ${(9 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

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

Step 2.1.1.18

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

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

Step 2.1.1.19

Combine the numerators over the common denominator.

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

Step 2.1.1.20

Multiply ${(9 - x)}^{\frac{1}{2}}$ by ${(9 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1.1.20.1

Move ${(9 - x)}^{\frac{1}{2}}$ .

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

Step 2.1.1.20.2

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

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

Step 2.1.1.20.3

Combine the numerators over the common denominator.

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

Step 2.1.1.20.4

Add $1$ and $1$ .

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

Step 2.1.1.20.5

Divide $2$ by $2$ .

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

Step 2.1.1.21

Simplify ${(9 - x)}^{1} \cdot 2$ .

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

Step 2.1.1.22

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

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

Step 2.1.1.23

Simplify.

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Step 2.1.1.23.1

Apply the distributive property.

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

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 $9$ .

$\frac{- x + 18 + 2 (- x)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- x + 18 - 2 x}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.2.2

Subtract $2 x$ from $- x$ .

$\frac{- 3 x + 18}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.3

Factor $3$ out of $- 3 x + 18$ .

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Step 2.1.1.23.3.1

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

$\frac{3 (- x) + 18}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.3.2

Factor $3$ out of $18$ .

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

Step 2.1.1.23.3.3

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

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

Step 2.1.1.23.4

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

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

Step 2.1.1.23.5

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{3 (- (x) - 1 (- 6))}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$\frac{3 (- (x - 6))}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.7

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$\frac{3 (- 1 (x - 6))}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2.1.1.23.8

Move the negative in front of the fraction.

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3 (x - 6)}{2 {(9 - x)}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [\frac{x - 6}{{(9 - x)}^{\frac{1}{2}}}]$ .

$- \frac{3}{2} \frac{d}{d x} [\frac{x - 6}{{(9 - x)}^{\frac{1}{2}}}]$

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 $f (x) = x - 6$ and $g (x) = {(9 - x)}^{\frac{1}{2}}$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{{({(9 - x)}^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.2.3.1

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{{(9 - x)}^{\frac{1}{2} \cdot 2}}$

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.

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{{(9 - x)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.2.3.2.2

Rewrite the expression.

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{{(9 - x)}^{1}}$

Step 2.1.2.4

Simplify.

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

Step 2.1.2.5

Differentiate.

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Step 2.1.2.5.1

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

Step 2.1.2.5.2

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (1 + \frac{d}{d x} [- 6]) - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

Step 2.1.2.5.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (1 + 0) - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

Step 2.1.2.5.4

Simplify the expression.

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Step 2.1.2.5.4.1

Add $1$ and $0$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \cdot 1 - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

Step 2.1.2.5.4.2

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]}{9 - x}$

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 $g (x) = 9 - x$ .

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Step 2.1.2.6.1

To apply the Chain Rule, set $u_{1}$ as $9 - x$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [9 - x])}{9 - x}$

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 $n = \frac{1}{2}$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.6.3

Replace all occurrences of $u_{1}$ with $9 - x$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.7

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.8

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.9

Combine the numerators over the common denominator.

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.10

Simplify the numerator.

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Step 2.1.2.10.1

Multiply $- 1$ by $2$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.10.2

Subtract $2$ from $1$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{\frac{- 1}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

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.

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {(9 - x)}^{- \frac{1}{2}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.11.2

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{{(9 - x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.11.3

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \frac{d}{d x} [9 - x])}{9 - x}$

Step 2.1.2.12

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [9] + \frac{d}{d x} [- x]))}{9 - x}$

Step 2.1.2.13

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x]))}{9 - x}$

Step 2.1.2.14

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \frac{d}{d x} [- x])}{9 - x}$

Step 2.1.2.15

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

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} (- \frac{d}{d x} [x]))}{9 - x}$

Step 2.1.2.16

Multiply.

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Step 2.1.2.16.1

Multiply $- 1$ by $- 1$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + 1 (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [x]))}{9 - x}$

Step 2.1.2.16.2

Multiply $x - 6$ by $1$ .

$- \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [x]))}{9 - x}$

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 $n = 1$ .

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

Step 2.1.2.18

Combine fractions.

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Step 2.1.2.18.1

Multiply $x - 6$ by $1$ .

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

Step 2.1.2.18.2

Multiply $\frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}}{9 - x}$ by $\frac{3}{2}$ .

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

Step 2.1.2.18.3

Reorder.

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Step 2.1.2.18.3.1

Move $3$ to the left of ${(9 - x)}^{\frac{1}{2}} + (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ .

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

Step 2.1.2.18.3.2

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

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

Step 2.1.2.19

Simplify.

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Step 2.1.2.19.1

Apply the distributive property.

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

Step 2.1.2.19.2

Apply the distributive property.

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

Step 2.1.2.19.3

Simplify the numerator.

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Step 2.1.2.19.3.1

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

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Step 2.1.2.19.3.1.1

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

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

Step 2.1.2.19.3.1.2

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

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

Step 2.1.2.19.3.1.3

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

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

Step 2.1.2.19.3.2

Let $u_{2} = {(9 - x)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(9 - x)}^{\frac{1}{2}}$ .

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Step 2.1.2.19.3.2.1

Rewrite using the commutative property of multiplication.

$- \frac{3 \frac{2 u_{2} \cdot u_{2} + x - 6}{2 u_{2}}}{2 \cdot 9 + 2 (- x)}$

Step 2.1.2.19.3.2.2

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

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Step 2.1.2.19.3.2.2.1

Move $u_{2}$ .

$- \frac{3 \frac{2 (u_{2} \cdot u_{2}) + x - 6}{2 u_{2}}}{2 \cdot 9 + 2 (- x)}$

Step 2.1.2.19.3.2.2.2

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

$- \frac{3 \frac{2 {u_{2}}^{2} + x - 6}{2 u_{2}}}{2 \cdot 9 + 2 (- x)}$

Step 2.1.2.19.3.3

Replace all occurrences of $u_{2}$ with ${(9 - x)}^{\frac{1}{2}}$ .

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

Step 2.1.2.19.3.4

Simplify.

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Step 2.1.2.19.3.4.1

Simplify each term.

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Step 2.1.2.19.3.4.1.1

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.2.19.3.4.1.1.1

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

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

Step 2.1.2.19.3.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.19.3.4.1.1.2.1

Cancel the common factor.

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

Step 2.1.2.19.3.4.1.1.2.2

Rewrite the expression.

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

Step 2.1.2.19.3.4.1.2

Simplify.

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

Step 2.1.2.19.3.4.1.3

Apply the distributive property.

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

Step 2.1.2.19.3.4.1.4

Multiply $2$ by $9$ .

$- \frac{3 \frac{18 + 2 (- x) + x - 6}{2 {(9 - x)}^{\frac{1}{2}}}}{2 \cdot 9 + 2 (- x)}$

Step 2.1.2.19.3.4.1.5

Multiply $- 1$ by $2$ .

$- \frac{3 \frac{18 - 2 x + x - 6}{2 {(9 - x)}^{\frac{1}{2}}}}{2 \cdot 9 + 2 (- x)}$

Step 2.1.2.19.3.4.2

Subtract $6$ from $18$ .

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

Step 2.1.2.19.3.4.3

Add $- 2 x$ and $x$ .

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

Step 2.1.2.19.4

Combine terms.

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Step 2.1.2.19.4.1

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

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

Step 2.1.2.19.4.2

Multiply $2$ by $9$ .

$- \frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 + 2 (- x)}$

Step 2.1.2.19.4.3

Multiply $- 1$ by $2$ .

$- \frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 - 2 x}$

Step 2.1.2.19.4.4

Rewrite $\frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 - 2 x}$ as a product.

$- (\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{1}{18 - 2 x})$

Step 2.1.2.19.4.5

Multiply $\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}$ by $\frac{1}{18 - 2 x}$ .

$- \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} (18 - 2 x)}$

Step 2.1.2.19.5

Simplify the denominator.

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Step 2.1.2.19.5.1

Factor $2$ out of $18 - 2 x$ .

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Step 2.1.2.19.5.1.1

Factor $2$ out of $18$ .

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

Step 2.1.2.19.5.1.2

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

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

Step 2.1.2.19.5.1.3

Factor $2$ out of $2 (9) + 2 (- x)$ .

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

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

Step 2.1.2.19.5.2

Combine exponents.

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Step 2.1.2.19.5.2.1

Multiply $2$ by $2$ .

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

Step 2.1.2.19.5.2.2

Raise $9 - x$ to the power of $1$ .

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

Step 2.1.2.19.5.2.3

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

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

Step 2.1.2.19.5.2.4

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

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

Step 2.1.2.19.5.2.5

Combine the numerators over the common denominator.

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

Step 2.1.2.19.5.2.6

Add $2$ and $1$ .

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

Step 2.1.2.19.6

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

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

Step 2.1.2.19.7

Rewrite $12$ as $- 1 (- 12)$ .

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

Step 2.1.2.19.8

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

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

Step 2.1.2.19.9

Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

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

Step 2.1.2.19.10

Move the negative in front of the fraction.

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

Step 2.1.2.19.11

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.19.12

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

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

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$ .

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

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$3 (x - 12) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

Divide each term in $3 (x - 12) = 0$ by $3$ and simplify.

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Step 2.2.3.1.1

Divide each term in $3 (x - 12) = 0$ by $3$ .

$\frac{3 (x - 12)}{3} = \frac{0}{3}$

Step 2.2.3.1.2

Simplify the left side.

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Step 2.2.3.1.2.1

Cancel the common factor of $3$ .

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Step 2.2.3.1.2.1.1

Cancel the common factor.

$\frac{3 (x - 12)}{3} = \frac{0}{3}$

Step 2.2.3.1.2.1.2

Divide $x - 12$ by $1$ .

$x - 12 = \frac{0}{3}$

Step 2.2.3.1.3

Simplify the right side.

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Step 2.2.3.1.3.1

Divide $0$ by $3$ .

$x - 12 = 0$

Step 2.2.3.2

Add $12$ to both sides of the equation.

$x = 12$

Step 2.2.4

Exclude the solutions that do not make $\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}} = 0$ true.

$No solution$

Step 3

Find the domain of $f (x) = x \sqrt{9 - x}$ .

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Step 3.1

Set the radicand in $\sqrt{9 - x}$ greater than or equal to $0$ to find where the expression is defined.

$9 - x \geq 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $9$ from both sides of the inequality.

$- x \geq - 9$

Step 3.2.2

Divide each term in $- x \geq - 9$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- x \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 9}{- 1}$

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.

$\frac{x}{1} \leq \frac{- 9}{- 1}$

Step 3.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 9}{- 1}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 9$ by $- 1$ .

$x \leq 9$

Step 3.3

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

Interval Notation:

$(- \infty, 9]$

Set-Builder Notation:

${x | x \leq 9}$

Interval Notation:

$(- \infty, 9]$

Set-Builder Notation:

${x | x \leq 9}$

Step 4

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

$(- \infty, 9]$

Step 5

Substitute any number from the interval $(- \infty, 9]$ 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.

$f'' (0) = \frac{3 ((0) - 12)}{4 {(9 - (0))}^{\frac{3}{2}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Factor $4$ out of $3 ((0) - 12)$ .

$f'' (0) = \frac{4 (3 (0 - 3))}{4 {(9 - (0))}^{\frac{3}{2}}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $4$ out of $4 {(9 - (0))}^{\frac{3}{2}}$ .

$f'' (0) = \frac{4 (3 (0 - 3))}{4 ({(9 - (0))}^{\frac{3}{2}})}$

Step 5.2.2.2

Cancel the common factor.

$f'' (0) = \frac{4 (3 (0 - 3))}{4 {(9 - (0))}^{\frac{3}{2}}}$

Step 5.2.2.3

Rewrite the expression.

$f'' (0) = \frac{3 (0 - 3)}{{(9 - (0))}^{\frac{3}{2}}}$

Step 5.2.3

Subtract $3$ from $0$ .

$f'' (0) = \frac{3 \cdot - 3}{{(9 - (0))}^{\frac{3}{2}}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Subtract $0$ from $9$ .

$f'' (0) = \frac{3 \cdot - 3}{9^{\frac{3}{2}}}$

Step 5.2.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 5.2.4.3

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

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

Step 5.2.4.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.1

Cancel the common factor.

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

Step 5.2.4.4.2

Rewrite the expression.

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

Step 5.2.4.5

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

$f'' (0) = \frac{3 \cdot - 3}{27}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Multiply $3$ by $- 3$ .

$f'' (0) = \frac{- 9}{27}$

Step 5.2.5.2

Cancel the common factor of $- 9$ and $27$ .

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Step 5.2.5.2.1

Factor $9$ out of $- 9$ .

$f'' (0) = \frac{9 (- 1)}{27}$

Step 5.2.5.2.2

Cancel the common factors.

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Step 5.2.5.2.2.1

Factor $9$ out of $27$ .

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

Step 5.2.5.2.2.2

Cancel the common factor.

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

Step 5.2.5.2.2.3

Rewrite the expression.

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

Step 5.2.5.3

Move the negative in front of the fraction.

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

Step 5.2.6

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

$- \frac{1}{3}$

Step 5.3

The graph is concave down on the interval $(- \infty, 9]$ because $f'' (0)$ is negative.

Concave down on $(- \infty, 9]$ since $f'' (x)$ is negative

Step 6