Calculus Examples

$f (x) = \frac{x + 5}{\sqrt{x}}$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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 $f (x) = x + 5$ and $g (x) = x^{\frac{1}{2}}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

$\frac{x^{\frac{1}{2}} (1 + 0) - (x + 5) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 1.8

Simplify the expression.

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Step 1.8.1

Add $1$ and $0$ .

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

Step 1.8.2

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Combine the numerators over the common denominator.

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

Step 1.13

Simplify the numerator.

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Step 1.13.1

Multiply $- 1$ by $2$ .

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

Step 1.13.2

Subtract $2$ from $1$ .

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

Step 1.14

Move the negative in front of the fraction.

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

Simplify.

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Step 1.17.1

Apply the distributive property.

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

Step 1.17.2

Apply the distributive property.

$\frac{x^{\frac{1}{2}} - x \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3

Simplify the numerator.

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Step 1.17.3.1

Simplify each term.

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Step 1.17.3.1.1

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

$\frac{x^{\frac{1}{2}} - \frac{x}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.2

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

$\frac{x^{\frac{1}{2}} - \frac{x \cdot x^{- \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.3

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

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Step 1.17.3.1.3.1

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

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Step 1.17.3.1.3.1.1

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

$\frac{x^{\frac{1}{2}} - \frac{x^{1} x^{- \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.3.1.2

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

$\frac{x^{\frac{1}{2}} - \frac{x^{1 - \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.3.2

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

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.3.3

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{2 - 1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.3.4

Subtract $1$ from $2$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.4

Multiply $- 1$ by $5$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.5

Combine $- 5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} + \frac{- 5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.1.6

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.2

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

$\frac{x^{\frac{1}{2}} \cdot \frac{2}{2} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.3

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

$\frac{\frac{x^{\frac{1}{2}} \cdot 2}{2} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} \cdot 2 - x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.5

Subtract $x^{\frac{1}{2}}$ from $x^{\frac{1}{2}} \cdot 2$ .

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Step 1.17.3.5.1

Reorder $x^{\frac{1}{2}}$ and $2$ .

$\frac{\frac{2 \cdot x^{\frac{1}{2}} - x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.3.5.2

Subtract $x^{\frac{1}{2}}$ from $2 \cdot x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.4

Combine terms.

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Step 1.17.4.1

Multiply $\frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 1.17.4.2

Combine.

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

Step 1.17.4.3

Apply the distributive property.

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

Step 1.17.4.4

Cancel the common factor of $2$ .

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Step 1.17.4.4.1

Cancel the common factor.

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

Step 1.17.4.4.2

Rewrite the expression.

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

Step 1.17.4.5

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{1}{2}} - 2 \frac{5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.6

Combine $- 2$ and $\frac{5}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} + \frac{- 2 \cdot 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.7

Multiply $- 2$ by $5$ .

$\frac{x^{\frac{1}{2}} + \frac{- 10}{2 x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.8

Factor $2$ out of $- 10$ .

$\frac{x^{\frac{1}{2}} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.9

Cancel the common factors.

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Step 1.17.4.9.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 1.17.4.9.2

Cancel the common factor.

$\frac{x^{\frac{1}{2}} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.9.3

Rewrite the expression.

$\frac{x^{\frac{1}{2}} + \frac{- 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.4.10

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} - \frac{5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5

Simplify the numerator.

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Step 1.17.5.1

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

$\frac{\frac{x^{\frac{1}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.2

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} x^{\frac{1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.3

Simplify the numerator.

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Step 1.17.5.3.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.17.5.3.1.1

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

$\frac{\frac{x^{\frac{1}{2} + \frac{1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.3.1.2

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1 + 1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.3.1.3

Add $1$ and $1$ .

$\frac{\frac{x^{\frac{2}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.3.1.4

Divide $2$ by $2$ .

$\frac{\frac{x^{1} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.5.3.2

Simplify $x^{1}$ .

$\frac{\frac{x - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 1.17.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 5}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$

Step 1.17.7

Multiply $\frac{x - 5}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$ .

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Step 1.17.7.1

Multiply $\frac{x - 5}{x^{\frac{1}{2}}}$ by $\frac{1}{2 x}$ .

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

Step 1.17.7.2

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 1.17.7.2.1

Move $x$ .

$\frac{x - 5}{x \cdot x^{\frac{1}{2}} \cdot 2}$

Step 1.17.7.2.2

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

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Step 1.17.7.2.2.1

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

$\frac{x - 5}{x^{1} x^{\frac{1}{2}} \cdot 2}$

Step 1.17.7.2.2.2

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

$\frac{x - 5}{x^{1 + \frac{1}{2}} \cdot 2}$

Step 1.17.7.2.3

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

$\frac{x - 5}{x^{\frac{2}{2} + \frac{1}{2}} \cdot 2}$

Step 1.17.7.2.4

Combine the numerators over the common denominator.

$\frac{x - 5}{x^{\frac{2 + 1}{2}} \cdot 2}$

Step 1.17.7.2.5

Add $2$ and $1$ .

$\frac{x - 5}{x^{\frac{3}{2}} \cdot 2}$

Step 1.17.8

Move $2$ to the left of $x^{\frac{3}{2}}$ .

$\frac{x - 5}{2 x^{\frac{3}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x - 5}{2 x^{\frac{3}{2}}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{x - 5}{x^{\frac{3}{2}}}]$ .

$f'' (x) = \frac{1}{2} \cdot \frac{d}{d x} (\frac{x - 5}{x^{\frac{3}{2}}})$

Step 2.2

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} \frac{d}{d x} (x - 5) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{{(x^{\frac{3}{2}})}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

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Step 2.3.1.1

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} \frac{d}{d x} (x - 5) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{\frac{3}{2} \cdot 2}}$

Step 2.3.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.2.1

Cancel the common factor.

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} \frac{d}{d x} (x - 5) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{\frac{3}{2} \cdot 2}}$

Step 2.3.1.2.2

Rewrite the expression.

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} \frac{d}{d x} (x - 5) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.2

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5)) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.3

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} (1 + \frac{d}{d x} (- 5)) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.4

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} (1 + 0) - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} \cdot 1 - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.5.2

Multiply $x^{\frac{3}{2}}$ by $1$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) \frac{d}{d x} x^{\frac{3}{2}}}{x^{3}}$

Step 2.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{3}{2} - 1})}{x^{3}}$

Step 2.4

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})}{x^{3}}$

Step 2.5

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

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})}{x^{3}}$

Step 2.6

Combine the numerators over the common denominator.

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})}{x^{3}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{3 - 2}{2}})}{x^{3}}$

Step 2.7.2

Subtract $2$ from $3$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) (\frac{3}{2} x^{\frac{1}{2}})}{x^{3}}$

Step 2.8

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$f'' (x) = \frac{1}{2} \cdot \frac{x^{\frac{3}{2}} - (x - 5) \frac{3 x^{\frac{1}{2}}}{2}}{x^{3}}$

Step 2.9

Multiply $\frac{1}{2}$ by $\frac{x^{\frac{3}{2}} - (x - 5) \frac{3 x^{\frac{1}{2}}}{2}}{x^{3}}$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - (x - 5) \frac{3 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = \frac{x^{\frac{3}{2}} + (- x + 5) (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.2

Apply the distributive property.

$f'' (x) = \frac{x^{\frac{3}{2}} - x \frac{3 x^{\frac{1}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3

Simplify the numerator.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

Multiply $- x \frac{3 x^{\frac{1}{2}}}{2}$ .

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Step 2.10.3.1.1.1

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $x$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{1}{2}} x}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.1.2

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

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 (x \cdot x^{\frac{1}{2}})}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.1.3

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

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{1 + \frac{1}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.1.4

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

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{2}{2} + \frac{1}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.1.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{2 + 1}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.1.6

Add $2$ and $1$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{3}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.2

Multiply $- 1$ by $- 5$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{3}{2}}}{2} + 5 (\frac{3 x^{\frac{1}{2}}}{2})}{2 x^{3}}$

Step 2.10.3.1.3

Multiply $5 \frac{3 x^{\frac{1}{2}}}{2}$ .

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Step 2.10.3.1.3.1

Combine $5$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{3}{2}}}{2} + \frac{5 (3 x^{\frac{1}{2}})}{2}}{2 x^{3}}$

Step 2.10.3.1.3.2

Multiply $3$ by $5$ .

$f'' (x) = \frac{x^{\frac{3}{2}} - \frac{3 x^{\frac{3}{2}}}{2} + \frac{15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.2

To write $x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{x^{\frac{3}{2}} \cdot \frac{2}{2} - \frac{3 x^{\frac{3}{2}}}{2} + \frac{15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.3

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

$f'' (x) = \frac{\frac{x^{\frac{3}{2}} \cdot 2}{2} - \frac{3 x^{\frac{3}{2}}}{2} + \frac{15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.4

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{x^{\frac{3}{2}} \cdot 2 - 3 x^{\frac{3}{2}}}{2} + \frac{15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{x^{\frac{3}{2}} \cdot 2 - 3 x^{\frac{3}{2}} + 15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.6

Move $2$ to the left of $x^{\frac{3}{2}}$ .

$f'' (x) = \frac{\frac{2 x^{\frac{3}{2}} - 3 x^{\frac{3}{2}} + 15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.7

Subtract $3 x^{\frac{3}{2}}$ from $2 x^{\frac{3}{2}}$ .

$f'' (x) = \frac{\frac{- x^{\frac{3}{2}} + 15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.8

Simplify the numerator.

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Step 2.10.3.8.1

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{3}{2}} + 15 x^{\frac{1}{2}}$ .

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Step 2.10.3.8.1.1

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

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- x^{\frac{2}{2}}) + 15 x^{\frac{1}{2}}}{2}}{2 x^{3}}$

Step 2.10.3.8.1.2

Factor $x^{\frac{1}{2}}$ out of $15 x^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- x^{\frac{2}{2}}) + x^{\frac{1}{2}} \cdot 15}{2}}{2 x^{3}}$

Step 2.10.3.8.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- x^{\frac{2}{2}}) + x^{\frac{1}{2}} \cdot 15$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- x^{\frac{2}{2}} + 15)}{2}}{2 x^{3}}$

Step 2.10.3.8.2

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- x + 15)}{2}}{2 x^{3}}$

Step 2.10.3.8.3

Simplify.

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- x + 15)}{2}}{2 x^{3}}$

Step 2.10.3.9

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

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- (x) + 15)}{2}}{2 x^{3}}$

Step 2.10.3.10

Rewrite $15$ as $- 1 (- 15)$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- (x) - 1 \cdot - 15)}{2}}{2 x^{3}}$

Step 2.10.3.11

Factor $- 1$ out of $- (x) - 1 (- 15)$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- (x - 15))}{2}}{2 x^{3}}$

Step 2.10.3.12

Rewrite $- (x - 15)$ as $- 1 (x - 15)$ .

$f'' (x) = \frac{\frac{x^{\frac{1}{2}} (- 1 (x - 15))}{2}}{2 x^{3}}$

Step 2.10.3.13

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{x^{\frac{1}{2}} (x - 15)}{2}}{2 x^{3}}$

Step 2.10.4

Combine terms.

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Step 2.10.4.1

Rewrite $\frac{- \frac{x^{\frac{1}{2}} (x - 15)}{2}}{2 x^{3}}$ as a product.

$f'' (x) = - \frac{x^{\frac{1}{2}} (x - 15)}{2} \cdot \frac{1}{2 x^{3}}$

Step 2.10.4.2

Multiply $\frac{1}{2 x^{3}}$ by $\frac{x^{\frac{1}{2}} (x - 15)}{2}$ .

$f'' (x) = - \frac{x^{\frac{1}{2}} (x - 15)}{2 x^{3} \cdot 2}$

Step 2.10.4.3

Multiply $2$ by $2$ .

$f'' (x) = - \frac{x^{\frac{1}{2}} (x - 15)}{4 x^{3}}$

Step 2.10.4.4

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

$f'' (x) = - \frac{x - 15}{4 x^{3} x^{- \frac{1}{2}}}$

Step 2.10.4.5

Multiply $x^{3}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.10.4.5.1

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

$f'' (x) = - \frac{x - 15}{4 (x^{- \frac{1}{2}} x^{3})}$

Step 2.10.4.5.2

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

$f'' (x) = - \frac{x - 15}{4 x^{- \frac{1}{2} + 3}}$

Step 2.10.4.5.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{x - 15}{4 x^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Step 2.10.4.5.4

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

$f'' (x) = - \frac{x - 15}{4 x^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Step 2.10.4.5.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{x - 15}{4 x^{\frac{- 1 + 3 \cdot 2}{2}}}$

Step 2.10.4.5.6

Simplify the numerator.

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Step 2.10.4.5.6.1

Multiply $3$ by $2$ .

$f'' (x) = - \frac{x - 15}{4 x^{\frac{- 1 + 6}{2}}}$

Step 2.10.4.5.6.2

Add $- 1$ and $6$ .

$f'' (x) = - \frac{x - 15}{4 x^{\frac{5}{2}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{x - 5}{2 x^{\frac{3}{2}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

Cancel the common factor of $2$ .

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Step 4.1.3.2.1

Cancel the common factor.

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

Step 4.1.3.2.2

Rewrite the expression.

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

Step 4.1.4

Simplify.

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

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

$\frac{x^{\frac{1}{2}} (1 + 0) - (x + 5) \frac{d}{d x} [x^{\frac{1}{2}}]}{x}$

Step 4.1.8

Simplify the expression.

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Step 4.1.8.1

Add $1$ and $0$ .

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

Step 4.1.8.2

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

Combine the numerators over the common denominator.

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

Step 4.1.13

Simplify the numerator.

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Step 4.1.13.1

Multiply $- 1$ by $2$ .

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

Step 4.1.13.2

Subtract $2$ from $1$ .

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

Step 4.1.14

Move the negative in front of the fraction.

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

Step 4.1.15

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

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

Step 4.1.16

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

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

Step 4.1.17

Simplify.

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Step 4.1.17.1

Apply the distributive property.

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

Step 4.1.17.2

Apply the distributive property.

$\frac{x^{\frac{1}{2}} - x \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3

Simplify the numerator.

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Step 4.1.17.3.1

Simplify each term.

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Step 4.1.17.3.1.1

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

$\frac{x^{\frac{1}{2}} - \frac{x}{2 x^{\frac{1}{2}}} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.2

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

$\frac{x^{\frac{1}{2}} - \frac{x \cdot x^{- \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.3

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

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Step 4.1.17.3.1.3.1

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

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Step 4.1.17.3.1.3.1.1

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

$\frac{x^{\frac{1}{2}} - \frac{x^{1} x^{- \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.3.1.2

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

$\frac{x^{\frac{1}{2}} - \frac{x^{1 - \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.3.2

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

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.3.3

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{2 - 1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.3.4

Subtract $1$ from $2$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - 1 \cdot 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.4

Multiply $- 1$ by $5$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.5

Combine $- 5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} + \frac{- 5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.1.6

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.2

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

$\frac{x^{\frac{1}{2}} \cdot \frac{2}{2} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.3

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

$\frac{\frac{x^{\frac{1}{2}} \cdot 2}{2} - \frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} \cdot 2 - x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.5

Subtract $x^{\frac{1}{2}}$ from $x^{\frac{1}{2}} \cdot 2$ .

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Step 4.1.17.3.5.1

Reorder $x^{\frac{1}{2}}$ and $2$ .

$\frac{\frac{2 \cdot x^{\frac{1}{2}} - x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.3.5.2

Subtract $x^{\frac{1}{2}}$ from $2 \cdot x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.4

Combine terms.

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Step 4.1.17.4.1

Multiply $\frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}}}{x}$

Step 4.1.17.4.2

Combine.

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

Step 4.1.17.4.3

Apply the distributive property.

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

Step 4.1.17.4.4

Cancel the common factor of $2$ .

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Step 4.1.17.4.4.1

Cancel the common factor.

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

Step 4.1.17.4.4.2

Rewrite the expression.

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

Step 4.1.17.4.5

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{1}{2}} - 2 \frac{5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.6

Combine $- 2$ and $\frac{5}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} + \frac{- 2 \cdot 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.7

Multiply $- 2$ by $5$ .

$\frac{x^{\frac{1}{2}} + \frac{- 10}{2 x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.8

Factor $2$ out of $- 10$ .

$\frac{x^{\frac{1}{2}} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.9

Cancel the common factors.

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Step 4.1.17.4.9.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 4.1.17.4.9.2

Cancel the common factor.

$\frac{x^{\frac{1}{2}} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.9.3

Rewrite the expression.

$\frac{x^{\frac{1}{2}} + \frac{- 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.4.10

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} - \frac{5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5

Simplify the numerator.

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Step 4.1.17.5.1

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

$\frac{\frac{x^{\frac{1}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.2

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} x^{\frac{1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.3

Simplify the numerator.

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Step 4.1.17.5.3.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.17.5.3.1.1

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

$\frac{\frac{x^{\frac{1}{2} + \frac{1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.3.1.2

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1 + 1}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.3.1.3

Add $1$ and $1$ .

$\frac{\frac{x^{\frac{2}{2}} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.3.1.4

Divide $2$ by $2$ .

$\frac{\frac{x^{1} - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.5.3.2

Simplify $x^{1}$ .

$\frac{\frac{x - 5}{x^{\frac{1}{2}}}}{2 x}$

Step 4.1.17.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 5}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$

Step 4.1.17.7

Multiply $\frac{x - 5}{x^{\frac{1}{2}}} \cdot \frac{1}{2 x}$ .

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Step 4.1.17.7.1

Multiply $\frac{x - 5}{x^{\frac{1}{2}}}$ by $\frac{1}{2 x}$ .

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

Step 4.1.17.7.2

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 4.1.17.7.2.1

Move $x$ .

$\frac{x - 5}{x \cdot x^{\frac{1}{2}} \cdot 2}$

Step 4.1.17.7.2.2

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

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Step 4.1.17.7.2.2.1

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

$\frac{x - 5}{x^{1} x^{\frac{1}{2}} \cdot 2}$

Step 4.1.17.7.2.2.2

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

$\frac{x - 5}{x^{1 + \frac{1}{2}} \cdot 2}$

Step 4.1.17.7.2.3

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

$\frac{x - 5}{x^{\frac{2}{2} + \frac{1}{2}} \cdot 2}$

Step 4.1.17.7.2.4

Combine the numerators over the common denominator.

$\frac{x - 5}{x^{\frac{2 + 1}{2}} \cdot 2}$

Step 4.1.17.7.2.5

Add $2$ and $1$ .

$\frac{x - 5}{x^{\frac{3}{2}} \cdot 2}$

Step 4.1.17.8

Move $2$ to the left of $x^{\frac{3}{2}}$ .

$f' (x) = \frac{x - 5}{2 x^{\frac{3}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x - 5}{2 x^{\frac{3}{2}}}$ .

$\frac{x - 5}{2 x^{\frac{3}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x - 5}{2 x^{\frac{3}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x - 5}{2 x^{\frac{3}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$x - 5 = 0$

Step 5.3

Add $5$ to both sides of the equation.

$x = 5$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$\frac{x - 5}{2 \sqrt{x^{3}}}$

Step 6.2

Set the denominator in $\frac{x - 5}{2 \sqrt{x^{3}}}$ equal to $0$ to find where the expression is undefined.

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

${(2 \sqrt{x^{3}})}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(2 x^{\frac{3}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

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

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

Step 6.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

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Step 6.3.2.2.1.3.1

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

$4 x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$4 x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$4 x^{3} = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$4 x^{3} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Divide each term in $4 x^{3} = 0$ by $4$ and simplify.

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Step 6.3.3.1.1

Divide each term in $4 x^{3} = 0$ by $4$ .

$\frac{4 x^{3}}{4} = \frac{0}{4}$

Step 6.3.3.1.2

Simplify the left side.

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Step 6.3.3.1.2.1

Cancel the common factor of $4$ .

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Step 6.3.3.1.2.1.1

Cancel the common factor.

$\frac{4 x^{3}}{4} = \frac{0}{4}$

Step 6.3.3.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{4}$

Step 6.3.3.1.3

Simplify the right side.

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Step 6.3.3.1.3.1

Divide $0$ by $4$ .

$x^{3} = 0$

Step 6.3.3.2

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

$x = \sqrt[3]{0}$

Step 6.3.3.3

Simplify $\sqrt[3]{0}$ .

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Step 6.3.3.3.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.3.3.3.2

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

$x = 0$

Step 6.4

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 6.5.2

Simplify the equation.

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Step 6.5.2.1

Simplify the left side.

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Step 6.5.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 6.5.2.2

Simplify the right side.

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Step 6.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 6.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 6.5.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.6

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = 5$

Step 8

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{(5) - 15}{4 {(5)}^{\frac{5}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Subtract $15$ from $5$ .

$- \frac{- 10}{4 \cdot 5^{\frac{5}{2}}}$

Step 9.2

Factor $2$ out of $- 10$ .

$- \frac{2 (- 5)}{4 \cdot 5^{\frac{5}{2}}}$

Step 9.3

Cancel the common factors.

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Step 9.3.1

Factor $2$ out of $4 \cdot 5^{\frac{5}{2}}$ .

$- \frac{2 (- 5)}{2 (2 \cdot 5^{\frac{5}{2}})}$

Step 9.3.2

Cancel the common factor.

$- \frac{2 \cdot - 5}{2 (2 \cdot 5^{\frac{5}{2}})}$

Step 9.3.3

Rewrite the expression.

$- \frac{- 5}{2 \cdot 5^{\frac{5}{2}}}$

Step 9.4

Move the negative in front of the fraction.

$- - \frac{5}{2 \cdot 5^{\frac{5}{2}}}$

Step 9.5

Move $5^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- - \frac{1}{2 \cdot 5^{\frac{5}{2}} \cdot 5^{- 1}}$

Step 9.6

Multiply $5^{\frac{5}{2}}$ by $5^{- 1}$ by adding the exponents.

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Step 9.6.1

Move $5^{- 1}$ .

$- - \frac{1}{2 \cdot (5^{- 1} \cdot 5^{\frac{5}{2}})}$

Step 9.6.2

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

$- - \frac{1}{2 \cdot 5^{- 1 + \frac{5}{2}}}$

Step 9.6.3

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

$- - \frac{1}{2 \cdot 5^{- 1 \cdot \frac{2}{2} + \frac{5}{2}}}$

Step 9.6.4

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

$- - \frac{1}{2 \cdot 5^{\frac{- 1 \cdot 2}{2} + \frac{5}{2}}}$

Step 9.6.5

Combine the numerators over the common denominator.

$- - \frac{1}{2 \cdot 5^{\frac{- 1 \cdot 2 + 5}{2}}}$

Step 9.6.6

Simplify the numerator.

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Step 9.6.6.1

Multiply $- 1$ by $2$ .

$- - \frac{1}{2 \cdot 5^{\frac{- 2 + 5}{2}}}$

Step 9.6.6.2

Add $- 2$ and $5$ .

$- - \frac{1}{2 \cdot 5^{\frac{3}{2}}}$

Step 9.7

Multiply $- - \frac{1}{2 \cdot 5^{\frac{3}{2}}}$ .

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Step 9.7.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{2 \cdot 5^{\frac{3}{2}}}$

Step 9.7.2

Multiply $\frac{1}{2 \cdot 5^{\frac{3}{2}}}$ by $1$ .

$\frac{1}{2 \cdot 5^{\frac{3}{2}}}$

Step 10

$x = 5$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 5$ is a local minimum

Step 11

Find the y-value when $x = 5$ .

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Step 11.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{(5) + 5}{\sqrt{5}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Remove parentheses.

$f (5) = \frac{(5) + 5}{\sqrt{5}}$

Step 11.2.2

Add $5$ and $5$ .

$f (5) = \frac{10}{\sqrt{5}}$

Step 11.2.3

Multiply $\frac{10}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f (5) = \frac{10}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 11.2.4

Combine and simplify the denominator.

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Step 11.2.4.1

Multiply $\frac{10}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f (5) = \frac{10 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 11.2.4.2

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

$f (5) = \frac{10 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 11.2.4.3

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

$f (5) = \frac{10 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 11.2.4.4

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

$f (5) = \frac{10 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 11.2.4.5

Add $1$ and $1$ .

$f (5) = \frac{10 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 11.2.4.6

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

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Step 11.2.4.6.1

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

$f (5) = \frac{10 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 11.2.4.6.2

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

$f (5) = \frac{10 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 11.2.4.6.3

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

$f (5) = \frac{10 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 11.2.4.6.4

Cancel the common factor of $2$ .

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Step 11.2.4.6.4.1

Cancel the common factor.

$f (5) = \frac{10 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 11.2.4.6.4.2

Rewrite the expression.

$f (5) = \frac{10 \sqrt{5}}{5}$

Step 11.2.4.6.5

Evaluate the exponent.

$f (5) = \frac{10 \sqrt{5}}{5}$

Step 11.2.5

Cancel the common factor of $10$ and $5$ .

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Step 11.2.5.1

Factor $5$ out of $10 \sqrt{5}$ .

$f (5) = \frac{5 (2 \sqrt{5})}{5}$

Step 11.2.5.2

Cancel the common factors.

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Step 11.2.5.2.1

Factor $5$ out of $5$ .

$f (5) = \frac{5 (2 \sqrt{5})}{5 (1)}$

Step 11.2.5.2.2

Cancel the common factor.

$f (5) = \frac{5 (2 \sqrt{5})}{5 \cdot 1}$

Step 11.2.5.2.3

Rewrite the expression.

$f (5) = \frac{2 \sqrt{5}}{1}$

Step 11.2.5.2.4

Divide $2 \sqrt{5}$ by $1$ .

$f (5) = 2 \sqrt{5}$

Step 11.2.6

The final answer is $2 \sqrt{5}$ .

$y = 2 \sqrt{5}$

Step 12

These are the local extrema for $f (x) = \frac{x + 5}{\sqrt{x}}$ .

$(5, 2 \sqrt{5})$ is a local minima

Step 13