Calculus Examples

$f (x) = 9 \sqrt{x} - 2 x + 5$ , $[0, 6]$

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

By the Sum Rule, the derivative of $9 \sqrt{x} - 2 x + 5$ with respect to $x$ is $\frac{d}{d x} [9 \sqrt{x}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [9 \sqrt{x}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [9 \sqrt{x}]$ .

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Step 1.1.1.2.1

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

$\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.2

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

$9 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.3

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

$9 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.4

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

$9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.5

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

$9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.6

Combine the numerators over the common denominator.

$9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.7

Simplify the numerator.

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Step 1.1.1.2.7.1

Multiply $- 1$ by $2$ .

$9 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.7.2

Subtract $2$ from $1$ .

$9 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.8

Move the negative in front of the fraction.

$9 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.9

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

$9 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.10

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

$\frac{9 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.2.11

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

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 1.1.1.3.1

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

$\frac{9}{2 x^{\frac{1}{2}}} - 2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.1.3.2

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

$\frac{9}{2 x^{\frac{1}{2}}} - 2 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.1.3.3

Multiply $- 2$ by $1$ .

$\frac{9}{2 x^{\frac{1}{2}}} - 2 + \frac{d}{d x} [5]$

Step 1.1.1.4

Differentiate using the Constant Rule.

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Step 1.1.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 1.1.1.4.2

Add $\frac{9}{2 x^{\frac{1}{2}}} - 2$ and $0$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Add $2$ to both sides of the equation.

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

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 x^{\frac{1}{2}}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$2 x^{\frac{1}{2}}$

Step 1.2.4

Multiply each term in $\frac{9}{2 x^{\frac{1}{2}}} = 2$ by $2 x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{9}{2 x^{\frac{1}{2}}} = 2$ by $2 x^{\frac{1}{2}}$ .

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Cancel the common factor.

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

Step 1.2.4.2.2.2

Rewrite the expression.

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

Step 1.2.4.2.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 1.2.4.2.3.1

Cancel the common factor.

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

Step 1.2.4.2.3.2

Rewrite the expression.

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply $2$ by $2$ .

$9 = 4 x^{\frac{1}{2}}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $4 x^{\frac{1}{2}} = 9$ .

$4 x^{\frac{1}{2}} = 9$

Step 1.2.5.2

Divide each term in $4 x^{\frac{1}{2}} = 9$ by $4$ and simplify.

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Step 1.2.5.2.1

Divide each term in $4 x^{\frac{1}{2}} = 9$ by $4$ .

$\frac{4 x^{\frac{1}{2}}}{4} = \frac{9}{4}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor.

$\frac{4 x^{\frac{1}{2}}}{4} = \frac{9}{4}$

Step 1.2.5.2.2.2

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

$x^{\frac{1}{2}} = \frac{9}{4}$

Step 1.2.5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 1.2.5.4

Simplify the exponent.

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Step 1.2.5.4.1

Simplify the left side.

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Step 1.2.5.4.1.1

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

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Step 1.2.5.4.1.1.1

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

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Step 1.2.5.4.1.1.1.1

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

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

Step 1.2.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 1.2.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 1.2.5.4.1.1.2

Simplify.

$x = {(\frac{9}{4})}^{2}$

Step 1.2.5.4.2

Simplify the right side.

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Step 1.2.5.4.2.1

Simplify ${(\frac{9}{4})}^{2}$ .

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Step 1.2.5.4.2.1.1

Apply the product rule to $\frac{9}{4}$ .

$x = \frac{9^{2}}{4^{2}}$

Step 1.2.5.4.2.1.2

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

$x = \frac{81}{4^{2}}$

Step 1.2.5.4.2.1.3

Raise $4$ to the power of $2$ .

$x = \frac{81}{16}$

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.

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

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

Step 1.3.2

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

$2 \sqrt{x} = 0$

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.

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

Apply the product rule to $2 x^{\frac{1}{2}}$ .

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

Step 1.3.3.2.2.1.2

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

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

Step 1.3.3.2.2.1.3

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

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Step 1.3.3.2.2.1.3.1

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

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

Step 1.3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$4 x^{1} = 0^{2}$

Step 1.3.3.2.2.1.4

Simplify.

$4 x = 0^{2}$

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 $0$ .

$4 x = 0$

Step 1.3.3.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 1.3.3.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 1.3.4

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

$x < 0$

Step 1.3.5

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 1.4

Evaluate $9 \sqrt{x} - 2 x + 5$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{81}{16}$ .

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Step 1.4.1.1

Substitute $\frac{81}{16}$ for $x$ .

$9 \sqrt{\frac{81}{16}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$9 \sqrt{\frac{81}{16}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2

Simplify each term.

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Step 1.4.1.2.2.1

Rewrite $\sqrt{\frac{81}{16}}$ as $\frac{\sqrt{81}}{\sqrt{16}}$ .

$9 \frac{\sqrt{81}}{\sqrt{16}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.2

Simplify the numerator.

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Step 1.4.1.2.2.2.1

Rewrite $81$ as $9^{2}$ .

$9 \frac{\sqrt{9^{2}}}{\sqrt{16}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.2.2

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

$9 \frac{9}{\sqrt{16}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.3

Simplify the denominator.

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Step 1.4.1.2.2.3.1

Rewrite $16$ as $4^{2}$ .

$9 \frac{9}{\sqrt{4^{2}}} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.3.2

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

$9 (\frac{9}{4}) - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.4

Multiply $9 (\frac{9}{4})$ .

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Step 1.4.1.2.2.4.1

Combine $9$ and $\frac{9}{4}$ .

$\frac{9 \cdot 9}{4} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.4.2

Multiply $9$ by $9$ .

$\frac{81}{4} - 2 (\frac{81}{16}) + 5$

Step 1.4.1.2.2.5

Cancel the common factor of $2$ .

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Step 1.4.1.2.2.5.1

Factor $2$ out of $- 2$ .

$\frac{81}{4} + 2 (- 1) \frac{81}{16} + 5$

Step 1.4.1.2.2.5.2

Factor $2$ out of $16$ .

$\frac{81}{4} + 2 \cdot - 1 \frac{81}{2 \cdot 8} + 5$

Step 1.4.1.2.2.5.3

Cancel the common factor.

$\frac{81}{4} + 2 \cdot - 1 \frac{81}{2 \cdot 8} + 5$

Step 1.4.1.2.2.5.4

Rewrite the expression.

$\frac{81}{4} - 1 (\frac{81}{8}) + 5$

Step 1.4.1.2.2.6

Rewrite $- 1 (\frac{81}{8})$ as $- (\frac{81}{8})$ .

$\frac{81}{4} - \frac{81}{8} + 5$

Step 1.4.1.2.3

Find the common denominator.

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Step 1.4.1.2.3.1

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$\frac{81}{4} \cdot \frac{2}{2} - \frac{81}{8} + 5$

Step 1.4.1.2.3.2

Multiply $\frac{81}{4}$ by $\frac{2}{2}$ .

$\frac{81 \cdot 2}{4 \cdot 2} - \frac{81}{8} + 5$

Step 1.4.1.2.3.3

Write $5$ as a fraction with denominator $1$ .

$\frac{81 \cdot 2}{4 \cdot 2} - \frac{81}{8} + \frac{5}{1}$

Step 1.4.1.2.3.4

Multiply $\frac{5}{1}$ by $\frac{8}{8}$ .

$\frac{81 \cdot 2}{4 \cdot 2} - \frac{81}{8} + \frac{5}{1} \cdot \frac{8}{8}$

Step 1.4.1.2.3.5

Multiply $\frac{5}{1}$ by $\frac{8}{8}$ .

$\frac{81 \cdot 2}{4 \cdot 2} - \frac{81}{8} + \frac{5 \cdot 8}{8}$

Step 1.4.1.2.3.6

Reorder the factors of $4 \cdot 2$ .

$\frac{81 \cdot 2}{2 \cdot 4} - \frac{81}{8} + \frac{5 \cdot 8}{8}$

Step 1.4.1.2.3.7

Multiply $2$ by $4$ .

$\frac{81 \cdot 2}{8} - \frac{81}{8} + \frac{5 \cdot 8}{8}$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{81 \cdot 2 - 81 + 5 \cdot 8}{8}$

Step 1.4.1.2.5

Simplify each term.

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Step 1.4.1.2.5.1

Multiply $81$ by $2$ .

$\frac{162 - 81 + 5 \cdot 8}{8}$

Step 1.4.1.2.5.2

Multiply $5$ by $8$ .

$\frac{162 - 81 + 40}{8}$

Step 1.4.1.2.6

Simplify by adding and subtracting.

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Step 1.4.1.2.6.1

Subtract $81$ from $162$ .

$\frac{81 + 40}{8}$

Step 1.4.1.2.6.2

Add $81$ and $40$ .

$\frac{121}{8}$

Step 1.4.2

Evaluate at $x = 0$ .

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Step 1.4.2.1

Substitute $0$ for $x$ .

$9 \sqrt{0} - 2 \cdot 0 + 5$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Remove parentheses.

$9 \sqrt{0} - 2 \cdot 0 + 5$

Step 1.4.2.2.2

Simplify each term.

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Step 1.4.2.2.2.1

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

$9 \sqrt{0^{2}} - 2 \cdot 0 + 5$

Step 1.4.2.2.2.2

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

$9 \cdot 0 - 2 \cdot 0 + 5$

Step 1.4.2.2.2.3

Multiply $9$ by $0$ .

$0 - 2 \cdot 0 + 5$

Step 1.4.2.2.2.4

Multiply $- 2$ by $0$ .

$0 + 0 + 5$

Step 1.4.2.2.3

Simplify by adding numbers.

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Step 1.4.2.2.3.1

Add $0$ and $0$ .

$0 + 5$

Step 1.4.2.2.3.2

Add $0$ and $5$ .

$5$

Step 1.4.3

List all of the points.

$(\frac{81}{16}, \frac{121}{8}), (0, 5)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

$9 \sqrt{0} - 2 \cdot 0 + 5$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Remove parentheses.

$9 \sqrt{0} - 2 \cdot 0 + 5$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

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

$9 \sqrt{0^{2}} - 2 \cdot 0 + 5$

Step 2.1.2.2.2

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

$9 \cdot 0 - 2 \cdot 0 + 5$

Step 2.1.2.2.3

Multiply $9$ by $0$ .

$0 - 2 \cdot 0 + 5$

Step 2.1.2.2.4

Multiply $- 2$ by $0$ .

$0 + 0 + 5$

Step 2.1.2.3

Simplify by adding numbers.

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Step 2.1.2.3.1

Add $0$ and $0$ .

$0 + 5$

Step 2.1.2.3.2

Add $0$ and $5$ .

$5$

Step 2.2

Evaluate at $x = 6$ .

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Step 2.2.1

Substitute $6$ for $x$ .

$9 \sqrt{6} - 2 \cdot 6 + 5$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Remove parentheses.

$9 \sqrt{6} - 2 \cdot 6 + 5$

Step 2.2.2.2

Multiply $- 2$ by $6$ .

$9 \sqrt{6} - 12 + 5$

Step 2.2.2.3

Add $- 12$ and $5$ .

$9 \sqrt{6} - 7$

Step 2.3

List all of the points.

$(0, 5), (6, 9 \sqrt{6} - 7)$

Step 3

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: $(\frac{81}{16}, \frac{121}{8})$

Absolute Minimum: $(0, 5)$

Step 4