Calculus Examples

$y = x^{2} \sqrt{9 - x}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

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

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

Step 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 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

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

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

Step 1.1.13

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

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

Step 1.1.14

Combine fractions.

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Step 1.1.14.1

Multiply $- 1$ by $1$ .

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

Step 1.1.14.2

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

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

Step 1.1.14.3

Simplify the expression.

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Step 1.1.14.3.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 1.1.14.3.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 1.1.14.3.3

Move the negative in front of the fraction.

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

Step 1.1.15

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

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

Step 1.1.16

Reorder.

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Step 1.1.16.1

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

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

Step 1.1.16.2

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

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

Step 1.1.17

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

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

Step 1.1.18

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

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

Step 1.1.19

Combine the numerators over the common denominator.

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

Step 1.1.20

Multiply $2$ by $2$ .

$f' (x) = \frac{- x^{2} + 4 x {(- x + 9)}^{\frac{1}{2}} {(- x + 9)}^{\frac{1}{2}}}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.21

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

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Step 1.1.21.1

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

$f' (x) = \frac{- x^{2} + 4 x ({(- x + 9)}^{\frac{1}{2}} {(- x + 9)}^{\frac{1}{2}})}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.21.2

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

$f' (x) = \frac{- x^{2} + 4 x {(- x + 9)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.21.3

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + 4 x {(- x + 9)}^{\frac{1 + 1}{2}}}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.21.4

Add $1$ and $1$ .

$f' (x) = \frac{- x^{2} + 4 x {(- x + 9)}^{\frac{2}{2}}}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.21.5

Divide $2$ by $2$ .

$f' (x) = \frac{- x^{2} + 4 x (- x + 9)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.22

Simplify $4 x {(- x + 9)}^{1}$ .

$f' (x) = \frac{- x^{2} + 4 x (- x + 9)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23

Simplify.

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Step 1.1.23.1

Apply the distributive property.

$f' (x) = \frac{- x^{2} + 4 x (- x) + 4 x \cdot 9}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2

Simplify the numerator.

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Step 1.1.23.2.1

Simplify each term.

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Step 1.1.23.2.1.1

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{- x^{2} + 4 \cdot (- 1 x \cdot x) + 4 x \cdot 9}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2.1.2

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

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Step 1.1.23.2.1.2.1

Move $x$ .

$f' (x) = \frac{- x^{2} + 4 \cdot (- 1 (x \cdot x)) + 4 x \cdot 9}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2.1.2.2

Multiply $x$ by $x$ .

$f' (x) = \frac{- x^{2} + 4 \cdot (- 1 x^{2}) + 4 x \cdot 9}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2.1.3

Multiply $4$ by $- 1$ .

$f' (x) = \frac{- x^{2} - 4 x^{2} + 4 x \cdot 9}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2.1.4

Multiply $9$ by $4$ .

$f' (x) = \frac{- x^{2} - 4 x^{2} + 36 x}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.2.2

Subtract $4 x^{2}$ from $- x^{2}$ .

$f' (x) = \frac{- 5 x^{2} + 36 x}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.3

Factor $x$ out of $- 5 x^{2} + 36 x$ .

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Step 1.1.23.3.1

Factor $x$ out of $- 5 x^{2}$ .

$f' (x) = \frac{x (- 5 x) + 36 x}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.3.2

Factor $x$ out of $36 x$ .

$f' (x) = \frac{x (- 5 x) + x \cdot 36}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.3.3

Factor $x$ out of $x (- 5 x) + x \cdot 36$ .

$f' (x) = \frac{x (- 5 x + 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.4

Factor $- 1$ out of $- 5 x$ .

$f' (x) = \frac{x (- (5 x) + 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.5

Rewrite $36$ as $- 1 (- 36)$ .

$f' (x) = \frac{x (- (5 x) - 1 \cdot - 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.6

Factor $- 1$ out of $- (5 x) - 1 (- 36)$ .

$f' (x) = \frac{x (- (5 x - 36))}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.7

Rewrite $- (5 x - 36)$ as $- 1 (5 x - 36)$ .

$f' (x) = \frac{x (- 1 (5 x - 36))}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.1.23.8

Move the negative in front of the fraction.

$f' (x) = - \frac{x (5 x - 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x (5 x - 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$ .

$- \frac{x (5 x - 36)}{2 {(- x + 9)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x (5 x - 36)}{2 {(- x + 9)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x (5 x - 36)}{2 {(- x + 9)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$x (5 x - 36) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x = 0$

$5 x - 36 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $5 x - 36$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $5 x - 36$ equal to $0$ .

$5 x - 36 = 0$

Step 2.3.3.2

Solve $5 x - 36 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $36$ to both sides of the equation.

$5 x = 36$

Step 2.3.3.2.2

Divide each term in $5 x = 36$ by $5$ and simplify.

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Step 2.3.3.2.2.1

Divide each term in $5 x = 36$ by $5$ .

$\frac{5 x}{5} = \frac{36}{5}$

Step 2.3.3.2.2.2

Simplify the left side.

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Step 2.3.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{36}{5}$

Step 2.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{5}$

Step 2.3.4

The final solution is all the values that make $x (5 x - 36) = 0$ true.

$x = 0, \frac{36}{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$- \frac{x (5 x - 36)}{2 \sqrt{{(- x + 9)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{x (5 x - 36)}{2 \sqrt{- x + 9}}$

Step 3.2

Set the denominator in $\frac{x (5 x - 36)}{2 \sqrt{- x + 9}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

Apply the product rule to $2 {(- x + 9)}^{\frac{1}{2}}$ .

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

Step 3.3.2.2.1.2

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

$4 {({(- x + 9)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.3

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

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Step 3.3.2.2.1.3.1

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

$4 {(- x + 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(- x + 9)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(- x + 9)}^{1} = 0^{2}$

Step 3.3.2.2.1.4

Simplify.

$4 (- x + 9) = 0^{2}$

Step 3.3.2.2.1.5

Apply the distributive property.

$4 (- x) + 4 \cdot 9 = 0^{2}$

Step 3.3.2.2.1.6

Multiply.

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Step 3.3.2.2.1.6.1

Multiply $- 1$ by $4$ .

$- 4 x + 4 \cdot 9 = 0^{2}$

Step 3.3.2.2.1.6.2

Multiply $4$ by $9$ .

$- 4 x + 36 = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$- 4 x + 36 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $36$ from both sides of the equation.

$- 4 x = - 36$

Step 3.3.3.2

Divide each term in $- 4 x = - 36$ by $- 4$ and simplify.

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Step 3.3.3.2.1

Divide each term in $- 4 x = - 36$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 36}{- 4}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 36}{- 4}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 36}{- 4}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $- 36$ by $- 4$ .

$x = 9$

Step 3.4

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

$- x + 9 < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Subtract $9$ from both sides of the inequality.

$- x < - 9$

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2.2

Divide $x$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 9$ by $- 1$ .

$x > 9$

Step 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 $0$ .

$x \geq 9$

$[9, \infty)$

$x \geq 9$

$[9, \infty)$

Step 4

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

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{2} \sqrt{9 - (0)}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$0 \sqrt{9 - (0)}$

Step 4.1.2.2

Multiply $- 1$ by $0$ .

$0 \sqrt{9 + 0}$

Step 4.1.2.3

Add $9$ and $0$ .

$0 \sqrt{9}$

Step 4.1.2.4

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

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

Step 4.1.2.5

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

$0 \cdot 3$

Step 4.1.2.6

Multiply $0$ by $3$ .

$0$

Step 4.2

Evaluate at $x = \frac{36}{5}$ .

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Step 4.2.1

Substitute $\frac{36}{5}$ for $x$ .

${(\frac{36}{5})}^{2} \sqrt{9 - (\frac{36}{5})}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Apply the product rule to $\frac{36}{5}$ .

$\frac{36^{2}}{5^{2}} \sqrt{9 - (\frac{36}{5})}$

Step 4.2.2.2

Raise $36$ to the power of $2$ .

$\frac{1296}{5^{2}} \sqrt{9 - (\frac{36}{5})}$

Step 4.2.2.3

Raise $5$ to the power of $2$ .

$\frac{1296}{25} \sqrt{9 - (\frac{36}{5})}$

Step 4.2.2.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1296}{25} \sqrt{9 \cdot \frac{5}{5} - \frac{36}{5}}$

Step 4.2.2.5

Combine $9$ and $\frac{5}{5}$ .

$\frac{1296}{25} \sqrt{\frac{9 \cdot 5}{5} - \frac{36}{5}}$

Step 4.2.2.6

Combine the numerators over the common denominator.

$\frac{1296}{25} \sqrt{\frac{9 \cdot 5 - 36}{5}}$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $9$ by $5$ .

$\frac{1296}{25} \sqrt{\frac{45 - 36}{5}}$

Step 4.2.2.7.2

Subtract $36$ from $45$ .

$\frac{1296}{25} \sqrt{\frac{9}{5}}$

Step 4.2.2.8

Rewrite $\sqrt{\frac{9}{5}}$ as $\frac{\sqrt{9}}{\sqrt{5}}$ .

$\frac{1296}{25} \cdot \frac{\sqrt{9}}{\sqrt{5}}$

Step 4.2.2.9

Simplify the numerator.

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Step 4.2.2.9.1

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

$\frac{1296}{25} \cdot \frac{\sqrt{3^{2}}}{\sqrt{5}}$

Step 4.2.2.9.2

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

$\frac{1296}{25} \cdot \frac{3}{\sqrt{5}}$

Step 4.2.2.10

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1296}{25} (\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 4.2.2.11

Combine and simplify the denominator.

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Step 4.2.2.11.1

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.2.2.11.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 4.2.2.11.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 4.2.2.11.4

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

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.2.2.11.5

Add $1$ and $1$ .

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.2.2.11.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.2.2.11.6.2

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

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.2.2.11.6.3

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

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.2.2.11.6.4

Cancel the common factor of $2$ .

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Step 4.2.2.11.6.4.1

Cancel the common factor.

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.2.2.11.6.4.2

Rewrite the expression.

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5^{1}}$

Step 4.2.2.11.6.5

Evaluate the exponent.

$\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5}$

Step 4.2.2.12

Multiply $\frac{1296}{25} \cdot \frac{3 \sqrt{5}}{5}$ .

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Step 4.2.2.12.1

Multiply $\frac{1296}{25}$ by $\frac{3 \sqrt{5}}{5}$ .

$\frac{1296 (3 \sqrt{5})}{25 \cdot 5}$

Step 4.2.2.12.2

Multiply $3$ by $1296$ .

$\frac{3888 \sqrt{5}}{25 \cdot 5}$

Step 4.2.2.12.3

Multiply $25$ by $5$ .

$\frac{3888 \sqrt{5}}{125}$

Step 4.3

Evaluate at $x = 9$ .

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Step 4.3.1

Substitute $9$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Raise $9$ to the power of $2$ .

$81 \sqrt{9 - (9)}$

Step 4.3.2.2

Multiply $- 1$ by $9$ .

$81 \sqrt{9 - 9}$

Step 4.3.2.3

Subtract $9$ from $9$ .

$81 \sqrt{0}$

Step 4.3.2.4

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

$81 \sqrt{0^{2}}$

Step 4.3.2.5

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

$81 \cdot 0$

Step 4.3.2.6

Multiply $81$ by $0$ .

$0$

Step 4.4

List all of the points.

$(0, 0), (\frac{36}{5}, \frac{3888 \sqrt{5}}{125}), (9, 0)$

Step 5