Calculus Examples

$f (x) = x \sqrt{9 - x^{2}}$ , $[- 1, 3]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

To apply the Chain Rule, set $u$ as $9 - x^{2}$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Replace all occurrences of $u$ with $9 - x^{2}$ .

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.1.7

Simplify the numerator.

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Step 1.1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.1.8

Combine fractions.

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Step 1.1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.1.8.2

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

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

Step 1.1.1.8.3

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

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

Step 1.1.1.8.4

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

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

Step 1.1.1.9

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.10

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

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

Step 1.1.1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.12

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

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

Step 1.1.1.13

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

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

Step 1.1.1.14

Combine fractions.

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Step 1.1.1.14.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.14.2

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

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

Step 1.1.1.14.3

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

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

Step 1.1.1.15

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

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

Step 1.1.1.16

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

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

Step 1.1.1.17

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

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

Step 1.1.1.18

Add $1$ and $1$ .

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

Step 1.1.1.19

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

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

Step 1.1.1.20

Cancel the common factors.

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Step 1.1.1.20.1

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

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

Step 1.1.1.20.2

Cancel the common factor.

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

Step 1.1.1.20.3

Rewrite the expression.

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

Step 1.1.1.21

Move the negative in front of the fraction.

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

Step 1.1.1.22

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

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

Step 1.1.1.23

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

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

Step 1.1.1.24

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

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

Step 1.1.1.25

Combine the numerators over the common denominator.

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

Step 1.1.1.26

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

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Step 1.1.1.26.1

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

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

Step 1.1.1.26.2

Combine the numerators over the common denominator.

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

Step 1.1.1.26.3

Add $1$ and $1$ .

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

Step 1.1.1.26.4

Divide $2$ by $2$ .

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

Step 1.1.1.27

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

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

Step 1.1.1.28

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

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

Step 1.1.1.29

Reorder terms.

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 2 x^{2} + 9}{{(- x^{2} + 9)}^{\frac{1}{2}}}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 x^{2} + 9}{{(- x^{2} + 9)}^{\frac{1}{2}}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{- 2 x^{2} + 9}{{(- x^{2} + 9)}^{\frac{1}{2}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$- 2 x^{2} + 9 = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Subtract $9$ from both sides of the equation.

$- 2 x^{2} = - 9$

Step 1.2.3.2

Divide each term in $- 2 x^{2} = - 9$ by $- 2$ and simplify.

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Step 1.2.3.2.1

Divide each term in $- 2 x^{2} = - 9$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 9}{- 2}$

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 9}{- 2}$

Step 1.2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 9}{- 2}$

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{9}{2}$

Step 1.2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{9}{2}}$

Step 1.2.3.4

Simplify $\pm \sqrt{\frac{9}{2}}$ .

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Step 1.2.3.4.1

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{9}}{\sqrt{2}}$

Step 1.2.3.4.2

Simplify the numerator.

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Step 1.2.3.4.2.1

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

$x = \pm \frac{\sqrt{3^{2}}}{\sqrt{2}}$

Step 1.2.3.4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{3}{\sqrt{2}}$

Step 1.2.3.4.3

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2.3.4.4

Combine and simplify the denominator.

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Step 1.2.3.4.4.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.3.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.3.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.3.4.4.4

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

$x = \pm \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.3.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.3.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.3.4.4.6.1

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

$x = \pm \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.3.4.4.6.2

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

$x = \pm \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.4.4.6.3

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

$x = \pm \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.3.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{3 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.3.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{3 \sqrt{2}}{2^{1}}$

Step 1.2.3.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{3 \sqrt{2}}{2}$

Step 1.2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{3 \sqrt{2}}{2}$

Step 1.2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{3 \sqrt{2}}{2}$

Step 1.2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{2}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

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

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 9}{\sqrt{- x^{2} + 9}}$

Step 1.3.2

Set the denominator in $\frac{- 2 x^{2} + 9}{\sqrt{- x^{2} + 9}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 9} = 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

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

${\sqrt{- x^{2} + 9}}^{2} = 0^{2}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x^{2} + 9}$ as ${(- x^{2} + 9)}^{\frac{1}{2}}$ .

${({(- x^{2} + 9)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

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

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Step 1.3.3.2.2.1.1.1

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

${(- x^{2} + 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

${(- x^{2} + 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

${(- x^{2} + 9)}^{1} = 0^{2}$

Step 1.3.3.2.2.1.2

Simplify.

$- x^{2} + 9 = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

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

$- x^{2} + 9 = 0$

Step 1.3.3.3

Solve for $x$ .

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Step 1.3.3.3.1

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

Step 1.3.3.3.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 1.3.3.3.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 1.3.3.3.2.2

Simplify the left side.

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Step 1.3.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 1.3.3.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 9}{- 1}$

Step 1.3.3.3.2.3

Simplify the right side.

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Step 1.3.3.3.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 1.3.3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 1.3.3.3.4

Simplify $\pm \sqrt{9}$ .

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Step 1.3.3.3.4.1

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

$x = \pm \sqrt{3^{2}}$

Step 1.3.3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 1.3.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 1.3.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 1.3.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 1.3.4

Set the radicand in $\sqrt{- x^{2} + 9}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 9 < 0$

Step 1.3.5

Solve for $x$ .

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Step 1.3.5.1

Subtract $9$ from both sides of the inequality.

$- x^{2} < - 9$

Step 1.3.5.2

Divide each term in $- x^{2} < - 9$ by $- 1$ and simplify.

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Step 1.3.5.2.1

Divide each term in $- x^{2} < - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 9}{- 1}$

Step 1.3.5.2.2

Simplify the left side.

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Step 1.3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 9}{- 1}$

Step 1.3.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 9}{- 1}$

Step 1.3.5.2.3

Simplify the right side.

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Step 1.3.5.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} > 9$

Step 1.3.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{9}$

Step 1.3.5.4

Simplify the equation.

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Step 1.3.5.4.1

Simplify the left side.

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Step 1.3.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{9}$

Step 1.3.5.4.2

Simplify the right side.

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Step 1.3.5.4.2.1

Simplify $\sqrt{9}$ .

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Step 1.3.5.4.2.1.1

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

$| x | > \sqrt{3^{2}}$

Step 1.3.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 3 |$

Step 1.3.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | > 3$

Step 1.3.5.5

Write $| x | > 3$ as a piecewise.

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Step 1.3.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 1.3.5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 3$

Step 1.3.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 1.3.5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 3$

Step 1.3.5.5.5

Write as a piecewise.

${\begin{cases} x > 3 & x \geq 0 \\ - x > 3 & x < 0 \end{cases}$

Step 1.3.5.6

Find the intersection of $x > 3$ and $x \geq 0$ .

$x > 3$

Step 1.3.5.7

Divide each term in $- x > 3$ by $- 1$ and simplify.

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Step 1.3.5.7.1

Divide each term in $- x > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{3}{- 1}$

Step 1.3.5.7.2

Simplify the left side.

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Step 1.3.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{3}{- 1}$

Step 1.3.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{3}{- 1}$

Step 1.3.5.7.3

Simplify the right side.

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Step 1.3.5.7.3.1

Divide $3$ by $- 1$ .

$x < - 3$

Step 1.3.5.8

Find the union of the solutions.

$x < - 3$ or $x > 3$

Step 1.3.6

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 $0$ .

$x \leq - 3, x \geq 3$

$(- \infty, - 3] \cup [3, \infty)$

$x \leq - 3, x \geq 3$

$(- \infty, - 3] \cup [3, \infty)$

Step 1.4

Evaluate $x \sqrt{9 - x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{3 \sqrt{2}}{2}$ .

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Step 1.4.1.1

Substitute $\frac{3 \sqrt{2}}{2}$ for $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.1.2.1.1

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

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

Step 1.4.1.2.1.2

Apply the product rule to $3 \sqrt{2}$ .

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.1.2.2

Simplify the numerator.

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Step 1.4.1.2.2.1

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

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 {\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.1.2.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.1.2.2.2.1

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

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

Step 1.4.1.2.2.2.2

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

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 1.4.1.2.2.2.3

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

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.1.2.2.2.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.2.2.4.1

Cancel the common factor.

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.1.2.2.2.4.2

Rewrite the expression.

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{1}}{2^{2}}}$

Step 1.4.1.2.2.2.5

Evaluate the exponent.

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2}{2^{2}}}$

Step 1.4.1.2.3

Reduce the expression by cancelling the common factors.

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Step 1.4.1.2.3.1

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

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2}{4}}$

Step 1.4.1.2.3.2

Multiply $9$ by $2$ .

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{18}{4}}$

Step 1.4.1.2.3.3

Cancel the common factor of $18$ and $4$ .

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Step 1.4.1.2.3.3.1

Factor $2$ out of $18$ .

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

Step 1.4.1.2.3.3.2

Cancel the common factors.

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Step 1.4.1.2.3.3.2.1

Factor $2$ out of $4$ .

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 1.4.1.2.3.3.2.2

Cancel the common factor.

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 1.4.1.2.3.3.2.3

Rewrite the expression.

$\frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9}{2}}$

Step 1.4.1.2.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 \sqrt{2}}{2} \sqrt{9 \cdot \frac{2}{2} - \frac{9}{2}}$

Step 1.4.1.2.5

Combine $9$ and $\frac{2}{2}$ .

$\frac{3 \sqrt{2}}{2} \sqrt{\frac{9 \cdot 2}{2} - \frac{9}{2}}$

Step 1.4.1.2.6

Combine the numerators over the common denominator.

$\frac{3 \sqrt{2}}{2} \sqrt{\frac{9 \cdot 2 - 9}{2}}$

Step 1.4.1.2.7

Simplify the numerator.

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Step 1.4.1.2.7.1

Multiply $9$ by $2$ .

$\frac{3 \sqrt{2}}{2} \sqrt{\frac{18 - 9}{2}}$

Step 1.4.1.2.7.2

Subtract $9$ from $18$ .

$\frac{3 \sqrt{2}}{2} \sqrt{\frac{9}{2}}$

Step 1.4.1.2.8

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

$\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{9}}{\sqrt{2}}$

Step 1.4.1.2.9

Simplify the numerator.

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Step 1.4.1.2.9.1

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

$\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{3^{2}}}{\sqrt{2}}$

Step 1.4.1.2.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 \sqrt{2}}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.1.2.10

Simplify terms.

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Step 1.4.1.2.10.1

Cancel the common factor of $\sqrt{2}$ .

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Step 1.4.1.2.10.1.1

Factor $\sqrt{2}$ out of $3 \sqrt{2}$ .

$\frac{\sqrt{2} \cdot 3}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.1.2.10.1.2

Cancel the common factor.

$\frac{\sqrt{2} \cdot 3}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.1.2.10.1.3

Rewrite the expression.

$\frac{3}{2} \cdot 3$

Step 1.4.1.2.10.2

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

$\frac{3 \cdot 3}{2}$

Step 1.4.1.2.10.3

Multiply $3$ by $3$ .

$\frac{9}{2}$

Step 1.4.2

Evaluate at $x = - \frac{3 \sqrt{2}}{2}$ .

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Step 1.4.2.1

Substitute $- \frac{3 \sqrt{2}}{2}$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.2.2.1.1

Apply the product rule to $- \frac{3 \sqrt{2}}{2}$ .

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

Step 1.4.2.2.1.2

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

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

Step 1.4.2.2.1.3

Apply the product rule to $3 \sqrt{2}$ .

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

Step 1.4.2.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.4.2.2.2.1

Move ${(- 1)}^{2}$ .

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

Step 1.4.2.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.4.2.2.2.2.1

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

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

Step 1.4.2.2.2.2.2

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

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

Step 1.4.2.2.2.3

Add $2$ and $1$ .

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

Step 1.4.2.2.3

Raise $- 1$ to the power of $3$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{3^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.2.2.4

Simplify the numerator.

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Step 1.4.2.2.4.1

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

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 {\sqrt{2}}^{2}}{2^{2}}}$

Step 1.4.2.2.4.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.2.2.4.2.1

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

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

Step 1.4.2.2.4.2.2

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

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 1.4.2.2.4.2.3

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

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.2.2.4.2.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.4.2.4.1

Cancel the common factor.

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 1.4.2.2.4.2.4.2

Rewrite the expression.

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2^{1}}{2^{2}}}$

Step 1.4.2.2.4.2.5

Evaluate the exponent.

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2}{2^{2}}}$

Step 1.4.2.2.5

Reduce the expression by cancelling the common factors.

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Step 1.4.2.2.5.1

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

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9 \cdot 2}{4}}$

Step 1.4.2.2.5.2

Multiply $9$ by $2$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{18}{4}}$

Step 1.4.2.2.5.3

Cancel the common factor of $18$ and $4$ .

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Step 1.4.2.2.5.3.1

Factor $2$ out of $18$ .

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

Step 1.4.2.2.5.3.2

Cancel the common factors.

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Step 1.4.2.2.5.3.2.1

Factor $2$ out of $4$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 1.4.2.2.5.3.2.2

Cancel the common factor.

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{2 \cdot 9}{2 \cdot 2}}$

Step 1.4.2.2.5.3.2.3

Rewrite the expression.

$- \frac{3 \sqrt{2}}{2} \sqrt{9 - \frac{9}{2}}$

Step 1.4.2.2.6

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{9 \cdot \frac{2}{2} - \frac{9}{2}}$

Step 1.4.2.2.7

Combine $9$ and $\frac{2}{2}$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{\frac{9 \cdot 2}{2} - \frac{9}{2}}$

Step 1.4.2.2.8

Combine the numerators over the common denominator.

$- \frac{3 \sqrt{2}}{2} \sqrt{\frac{9 \cdot 2 - 9}{2}}$

Step 1.4.2.2.9

Simplify the numerator.

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Step 1.4.2.2.9.1

Multiply $9$ by $2$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{\frac{18 - 9}{2}}$

Step 1.4.2.2.9.2

Subtract $9$ from $18$ .

$- \frac{3 \sqrt{2}}{2} \sqrt{\frac{9}{2}}$

Step 1.4.2.2.10

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

$- \frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{9}}{\sqrt{2}}$

Step 1.4.2.2.11

Simplify the numerator.

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Step 1.4.2.2.11.1

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

$- \frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{3^{2}}}{\sqrt{2}}$

Step 1.4.2.2.11.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{3 \sqrt{2}}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.2.2.12

Simplify terms.

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Step 1.4.2.2.12.1

Cancel the common factor of $\sqrt{2}$ .

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Step 1.4.2.2.12.1.1

Move the leading negative in $- \frac{3 \sqrt{2}}{2}$ into the numerator.

$\frac{- 3 \sqrt{2}}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.2.2.12.1.2

Factor $\sqrt{2}$ out of $- 3 \sqrt{2}$ .

$\frac{\sqrt{2} \cdot - 3}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.2.2.12.1.3

Cancel the common factor.

$\frac{\sqrt{2} \cdot - 3}{2} \cdot \frac{3}{\sqrt{2}}$

Step 1.4.2.2.12.1.4

Rewrite the expression.

$\frac{- 3}{2} \cdot 3$

Step 1.4.2.2.12.2

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

$\frac{- 3 \cdot 3}{2}$

Step 1.4.2.2.12.3

Simplify the expression.

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Step 1.4.2.2.12.3.1

Multiply $- 3$ by $3$ .

$\frac{- 9}{2}$

Step 1.4.2.2.12.3.2

Move the negative in front of the fraction.

$- \frac{9}{2}$

Step 1.4.3

Evaluate at $x = - 3$ .

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Step 1.4.3.1

Substitute $- 3$ for $x$ .

$(- 3) \sqrt{9 - {(- 3)}^{2}}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

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

$- 3 \sqrt{9 - 1 \cdot 9}$

Step 1.4.3.2.2

Multiply $- 1$ by $9$ .

$- 3 \sqrt{9 - 9}$

Step 1.4.3.2.3

Subtract $9$ from $9$ .

$- 3 \sqrt{0}$

Step 1.4.3.2.4

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

$- 3 \sqrt{0^{2}}$

Step 1.4.3.2.5

Pull terms out from under the radical, assuming positive real numbers.

$- 3 \cdot 0$

Step 1.4.3.2.6

Multiply $- 3$ by $0$ .

$0$

Step 1.4.4

Evaluate at $x = 3$ .

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Step 1.4.4.1

Substitute $3$ for $x$ .

$(3) \sqrt{9 - {(3)}^{2}}$

Step 1.4.4.2

Simplify.

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Step 1.4.4.2.1

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

$3 \sqrt{9 - 1 \cdot 9}$

Step 1.4.4.2.2

Multiply $- 1$ by $9$ .

$3 \sqrt{9 - 9}$

Step 1.4.4.2.3

Subtract $9$ from $9$ .

$3 \sqrt{0}$

Step 1.4.4.2.4

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

$3 \sqrt{0^{2}}$

Step 1.4.4.2.5

Pull terms out from under the radical, assuming positive real numbers.

$3 \cdot 0$

Step 1.4.4.2.6

Multiply $3$ by $0$ .

$0$

Step 1.4.5

List all of the points.

$(\frac{3 \sqrt{2}}{2}, \frac{9}{2}), (- \frac{3 \sqrt{2}}{2}, - \frac{9}{2}), (- 3, 0), (3, 0)$

Step 2

Exclude the points that are not on the interval.

$(\frac{3 \sqrt{2}}{2}, \frac{9}{2}), (3, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 1$ .

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Step 3.1.1

Substitute $- 1$ for $x$ .

$(- 1) \sqrt{9 - {(- 1)}^{2}}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.1.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.1.2.1.1.1

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

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

Step 3.1.2.1.1.2

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

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

Step 3.1.2.1.2

Add $1$ and $2$ .

$- 1 \sqrt{9 + {(- 1)}^{3}}$

Step 3.1.2.2

Raise $- 1$ to the power of $3$ .

$- 1 \sqrt{9 - 1}$

Step 3.1.2.3

Subtract $1$ from $9$ .

$- 1 \sqrt{8}$

Step 3.1.2.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.2.4.1

Factor $4$ out of $8$ .

$- 1 \sqrt{4 (2)}$

Step 3.1.2.4.2

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

$- 1 \sqrt{2^{2} \cdot 2}$

Step 3.1.2.5

Pull terms out from under the radical.

$- 1 (2 \sqrt{2})$

Step 3.1.2.6

Multiply $2$ by $- 1$ .

$- 2 \sqrt{2}$

Step 3.2

Evaluate at $x = 3$ .

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Step 3.2.1

Substitute $3$ for $x$ .

$(3) \sqrt{9 - {(3)}^{2}}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$3 \sqrt{9 - 1 \cdot 9}$

Step 3.2.2.2

Multiply $- 1$ by $9$ .

$3 \sqrt{9 - 9}$

Step 3.2.2.3

Subtract $9$ from $9$ .

$3 \sqrt{0}$

Step 3.2.2.4

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

$3 \sqrt{0^{2}}$

Step 3.2.2.5

Pull terms out from under the radical, assuming positive real numbers.

$3 \cdot 0$

Step 3.2.2.6

Multiply $3$ by $0$ .

$0$

Step 3.3

List all of the points.

$(- 1, - 2 \sqrt{2}), (3, 0)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{3 \sqrt{2}}{2}, \frac{9}{2})$

Absolute Minimum: $(- 1, - 2 \sqrt{2})$

Step 5