Calculus Examples

$f (x) = x - 8 \sqrt{x - 5}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x - 8 \sqrt{x - 5}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8 \sqrt{x - 5}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 8 \sqrt{x - 5}]$

Step 1.1.2

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

$1 + \frac{d}{d x} [- 8 \sqrt{x - 5}]$

Step 1.2

Evaluate $\frac{d}{d x} [- 8 \sqrt{x - 5}]$ .

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Step 1.2.1

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

$1 + \frac{d}{d x} [- 8 {(x - 5)}^{\frac{1}{2}}]$

Step 1.2.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 {(x - 5)}^{\frac{1}{2}}$ with respect to $x$ is $- 8 \frac{d}{d x} [{(x - 5)}^{\frac{1}{2}}]$ .

$1 - 8 \frac{d}{d x} [{(x - 5)}^{\frac{1}{2}}]$

Step 1.2.3

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 $g (x) = x - 5$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$1 - 8 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 5])$

Step 1.2.3.2

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

$1 - 8 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5])$

Step 1.2.3.3

Replace all occurrences of $u$ with $x - 5$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5])$

Step 1.2.4

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]))$

Step 1.2.5

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [- 5]))$

Step 1.2.6

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + 0))$

Step 1.2.7

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0))$

Step 1.2.8

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0))$

Step 1.2.9

Combine the numerators over the common denominator.

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0))$

Step 1.2.10

Simplify the numerator.

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Step 1.2.10.1

Multiply $- 1$ by $2$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1 - 2}{2}} (1 + 0))$

Step 1.2.10.2

Subtract $2$ from $1$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{- 1}{2}} (1 + 0))$

Step 1.2.11

Move the negative in front of the fraction.

$1 - 8 (\frac{1}{2} {(x - 5)}^{- \frac{1}{2}} (1 + 0))$

Step 1.2.12

Add $1$ and $0$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{- \frac{1}{2}} \cdot 1)$

Step 1.2.13

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

$1 - 8 (\frac{{(x - 5)}^{- \frac{1}{2}}}{2} \cdot 1)$

Step 1.2.14

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

$1 - 8 \frac{{(x - 5)}^{- \frac{1}{2}}}{2}$

Step 1.2.15

Move ${(x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - 8 \frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.2.16

Combine $- 8$ and $\frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$ .

$1 + \frac{- 8}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.2.17

Factor $2$ out of $- 8$ .

$1 + \frac{2 \cdot - 4}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.2.18

Cancel the common factors.

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Step 1.2.18.1

Factor $2$ out of $2 {(x - 5)}^{\frac{1}{2}}$ .

$1 + \frac{2 \cdot - 4}{2 ({(x - 5)}^{\frac{1}{2}})}$

Step 1.2.18.2

Cancel the common factor.

$1 + \frac{2 \cdot - 4}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 1.2.18.3

Rewrite the expression.

$1 + \frac{- 4}{{(x - 5)}^{\frac{1}{2}}}$

Step 1.2.19

Move the negative in front of the fraction.

$1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{4}{{(x - 5)}^{\frac{1}{2}}}]$ .

$f'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- \frac{4}{{(x - 5)}^{\frac{1}{2}}})$

Step 2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \frac{4}{{(x - 5)}^{\frac{1}{2}}})$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{4}{{(x - 5)}^{\frac{1}{2}}}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{{(x - 5)}^{\frac{1}{2}}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{{(x - 5)}^{\frac{1}{2}}}]$ .

$f'' (x) = 0 - 4 \frac{d}{d x} \frac{1}{{(x - 5)}^{\frac{1}{2}}}$

Step 2.2.2

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

$f'' (x) = 0 - 4 \frac{d}{d x} {({(x - 5)}^{\frac{1}{2}})}^{- 1}$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = {(x - 5)}^{\frac{1}{2}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as ${(x - 5)}^{\frac{1}{2}}$ .

$f'' (x) = 0 - 4 (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} ({(x - 5)}^{\frac{1}{2}}))$

Step 2.2.3.2

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

$f'' (x) = 0 - 4 (- {u_{1}}^{- 2} \frac{d}{d x} {(x - 5)}^{\frac{1}{2}})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with ${(x - 5)}^{\frac{1}{2}}$ .

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} \frac{d}{d x} {(x - 5)}^{\frac{1}{2}})$

Step 2.2.4

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 $g (x) = x - 5$ .

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Step 2.2.4.1

To apply the Chain Rule, set $u_{2}$ as $x - 5$ .

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{d}{d u (2)} ({u_{2}}^{\frac{1}{2}}) \frac{d}{d x} (x - 5)))$

Step 2.2.4.2

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

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} (x - 5)))$

Step 2.2.4.3

Replace all occurrences of $u_{2}$ with $x - 5$ .

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot {(x - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} (x - 5)))$

Step 2.2.5

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

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} (x) + \frac{d}{d x} (- 5)))))$

Step 2.2.6

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

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} (- 5)))))$

Step 2.2.7

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

$f'' (x) = 0 - 4 (- {({(x - 5)}^{\frac{1}{2}})}^{- 2} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + 0))))$

Step 2.2.8

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{- 2}$ .

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Step 2.2.8.1

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

$f'' (x) = 0 - 4 (- {(x - 5)}^{\frac{1}{2} \cdot - 2} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + 0))))$

Step 2.2.8.2

Cancel the common factor of $2$ .

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Step 2.2.8.2.1

Factor $2$ out of $- 2$ .

$f'' (x) = 0 - 4 (- {(x - 5)}^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + 0))))$

Step 2.2.8.2.2

Cancel the common factor.

$f'' (x) = 0 - 4 (- {(x - 5)}^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + 0))))$

Step 2.2.8.2.3

Rewrite the expression.

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1} (1 + 0))))$

Step 2.2.9

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

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0))))$

Step 2.2.10

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

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0))))$

Step 2.2.11

Combine the numerators over the common denominator.

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0))))$

Step 2.2.12

Simplify the numerator.

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Step 2.2.12.1

Multiply $- 1$ by $2$ .

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{1 - 2}{2}} (1 + 0))))$

Step 2.2.12.2

Subtract $2$ from $1$ .

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{\frac{- 1}{2}} (1 + 0))))$

Step 2.2.13

Move the negative in front of the fraction.

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot ({(x - 5)}^{- \frac{1}{2}} (1 + 0))))$

Step 2.2.14

Add $1$ and $0$ .

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{1}{2} \cdot {(x - 5)}^{- \frac{1}{2}} \cdot 1))$

Step 2.2.15

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

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} (\frac{{(x - 5)}^{- \frac{1}{2}}}{2} \cdot 1))$

Step 2.2.16

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

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} \frac{{(x - 5)}^{- \frac{1}{2}}}{2})$

Step 2.2.17

Move ${(x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 0 - 4 (- {(x - 5)}^{- 1} \frac{1}{2 {(x - 5)}^{\frac{1}{2}}})$

Step 2.2.18

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

$f'' (x) = 0 - 4 (- \frac{{(x - 5)}^{- 1}}{2 {(x - 5)}^{\frac{1}{2}}})$

Step 2.2.19

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

$f'' (x) = 0 - 4 (- \frac{1}{2 {(x - 5)}^{\frac{1}{2}} (x - 5)})$

Step 2.2.20

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

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Step 2.2.20.1

Move $x - 5$ .

$f'' (x) = 0 - 4 (- \frac{1}{2 ((x - 5) {(x - 5)}^{\frac{1}{2}})})$

Step 2.2.20.2

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

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Step 2.2.20.2.1

Raise $x - 5$ to the power of $1$ .

$f'' (x) = 0 - 4 (- \frac{1}{2 ((x - 5) {(x - 5)}^{\frac{1}{2}})})$

Step 2.2.20.2.2

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

$f'' (x) = 0 - 4 (- \frac{1}{2 {(x - 5)}^{1 + \frac{1}{2}}})$

Step 2.2.20.3

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

$f'' (x) = 0 - 4 (- \frac{1}{2 {(x - 5)}^{\frac{2}{2} + \frac{1}{2}}})$

Step 2.2.20.4

Combine the numerators over the common denominator.

$f'' (x) = 0 - 4 (- \frac{1}{2 {(x - 5)}^{\frac{2 + 1}{2}}})$

Step 2.2.20.5

Add $2$ and $1$ .

$f'' (x) = 0 - 4 (- \frac{1}{2 {(x - 5)}^{\frac{3}{2}}})$

Step 2.2.21

Multiply $- 1$ by $- 4$ .

$f'' (x) = 0 + 4 (\frac{1}{2 {(x - 5)}^{\frac{3}{2}}})$

Step 2.2.22

Combine $4$ and $\frac{1}{2 {(x - 5)}^{\frac{3}{2}}}$ .

$f'' (x) = 0 + \frac{4}{2 {(x - 5)}^{\frac{3}{2}}}$

Step 2.2.23

Factor $2$ out of $4$ .

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

Step 2.2.24

Cancel the common factors.

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Step 2.2.24.1

Factor $2$ out of $2 {(x - 5)}^{\frac{3}{2}}$ .

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

Step 2.2.24.2

Cancel the common factor.

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

Step 2.2.24.3

Rewrite the expression.

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

Step 2.3

Add $0$ and $\frac{2}{{(x - 5)}^{\frac{3}{2}}}$ .

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

Step 3

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

$1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}} = 0$

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

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x - 8 \sqrt{x - 5}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8 \sqrt{x - 5}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 8 \sqrt{x - 5}]$

Step 4.1.1.2

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

$1 + \frac{d}{d x} [- 8 \sqrt{x - 5}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 8 \sqrt{x - 5}]$ .

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Step 4.1.2.1

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

$1 + \frac{d}{d x} [- 8 {(x - 5)}^{\frac{1}{2}}]$

Step 4.1.2.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 {(x - 5)}^{\frac{1}{2}}$ with respect to $x$ is $- 8 \frac{d}{d x} [{(x - 5)}^{\frac{1}{2}}]$ .

$1 - 8 \frac{d}{d x} [{(x - 5)}^{\frac{1}{2}}]$

Step 4.1.2.3

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 $g (x) = x - 5$ .

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Step 4.1.2.3.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$1 - 8 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - 5])$

Step 4.1.2.3.2

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

$1 - 8 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5])$

Step 4.1.2.3.3

Replace all occurrences of $u$ with $x - 5$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 5])$

Step 4.1.2.4

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]))$

Step 4.1.2.5

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [- 5]))$

Step 4.1.2.6

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1} (1 + 0))$

Step 4.1.2.7

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0))$

Step 4.1.2.8

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

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0))$

Step 4.1.2.9

Combine the numerators over the common denominator.

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0))$

Step 4.1.2.10

Simplify the numerator.

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Step 4.1.2.10.1

Multiply $- 1$ by $2$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{1 - 2}{2}} (1 + 0))$

Step 4.1.2.10.2

Subtract $2$ from $1$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{\frac{- 1}{2}} (1 + 0))$

Step 4.1.2.11

Move the negative in front of the fraction.

$1 - 8 (\frac{1}{2} {(x - 5)}^{- \frac{1}{2}} (1 + 0))$

Step 4.1.2.12

Add $1$ and $0$ .

$1 - 8 (\frac{1}{2} {(x - 5)}^{- \frac{1}{2}} \cdot 1)$

Step 4.1.2.13

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

$1 - 8 (\frac{{(x - 5)}^{- \frac{1}{2}}}{2} \cdot 1)$

Step 4.1.2.14

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

$1 - 8 \frac{{(x - 5)}^{- \frac{1}{2}}}{2}$

Step 4.1.2.15

Move ${(x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - 8 \frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 4.1.2.16

Combine $- 8$ and $\frac{1}{2 {(x - 5)}^{\frac{1}{2}}}$ .

$1 + \frac{- 8}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 4.1.2.17

Factor $2$ out of $- 8$ .

$1 + \frac{2 \cdot - 4}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 4.1.2.18

Cancel the common factors.

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Step 4.1.2.18.1

Factor $2$ out of $2 {(x - 5)}^{\frac{1}{2}}$ .

$1 + \frac{2 \cdot - 4}{2 ({(x - 5)}^{\frac{1}{2}})}$

Step 4.1.2.18.2

Cancel the common factor.

$1 + \frac{2 \cdot - 4}{2 {(x - 5)}^{\frac{1}{2}}}$

Step 4.1.2.18.3

Rewrite the expression.

$1 + \frac{- 4}{{(x - 5)}^{\frac{1}{2}}}$

Step 4.1.2.19

Move the negative in front of the fraction.

$f' (x) = 1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}}$ .

$1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$1 - \frac{4}{{(x - 5)}^{\frac{1}{2}}} = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$- \frac{4}{{(x - 5)}^{\frac{1}{2}}} = - 1$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

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

Step 5.3.2

The LCM of one and any expression is the expression.

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

Step 5.4

Multiply each term in $- \frac{4}{{(x - 5)}^{\frac{1}{2}}} = - 1$ by ${(x - 5)}^{\frac{1}{2}}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{4}{{(x - 5)}^{\frac{1}{2}}} = - 1$ by ${(x - 5)}^{\frac{1}{2}}$ .

$- \frac{4}{{(x - 5)}^{\frac{1}{2}}} {(x - 5)}^{\frac{1}{2}} = - {(x - 5)}^{\frac{1}{2}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of ${(x - 5)}^{\frac{1}{2}}$ .

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Step 5.4.2.1.1

Move the leading negative in $- \frac{4}{{(x - 5)}^{\frac{1}{2}}}$ into the numerator.

$\frac{- 4}{{(x - 5)}^{\frac{1}{2}}} {(x - 5)}^{\frac{1}{2}} = - {(x - 5)}^{\frac{1}{2}}$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- 4}{{(x - 5)}^{\frac{1}{2}}} {(x - 5)}^{\frac{1}{2}} = - {(x - 5)}^{\frac{1}{2}}$

Step 5.4.2.1.3

Rewrite the expression.

$- 4 = - {(x - 5)}^{\frac{1}{2}}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- {(x - 5)}^{\frac{1}{2}} = - 4$ .

$- {(x - 5)}^{\frac{1}{2}} = - 4$

Step 5.5.2

Divide each term in $- {(x - 5)}^{\frac{1}{2}} = - 4$ by $- 1$ and simplify.

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Step 5.5.2.1

Divide each term in $- {(x - 5)}^{\frac{1}{2}} = - 4$ by $- 1$ .

$\frac{- {(x - 5)}^{\frac{1}{2}}}{- 1} = \frac{- 4}{- 1}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 5)}^{\frac{1}{2}}}{1} = \frac{- 4}{- 1}$

Step 5.5.2.2.2

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

${(x - 5)}^{\frac{1}{2}} = \frac{- 4}{- 1}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Divide $- 4$ by $- 1$ .

${(x - 5)}^{\frac{1}{2}} = 4$

Step 5.5.3

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

${({(x - 5)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 5.5.4

Simplify the exponent.

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Step 5.5.4.1

Simplify the left side.

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Step 5.5.4.1.1

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

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Step 5.5.4.1.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 5.5.4.1.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 5.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.5.4.1.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 5.5.4.1.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = 4^{2}$

Step 5.5.4.1.1.2

Simplify.

$x - 5 = 4^{2}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

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

$x - 5 = 16$

Step 5.5.5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.5.5.1

Add $5$ to both sides of the equation.

$x = 16 + 5$

Step 5.5.5.2

Add $16$ and $5$ .

$x = 21$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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

$1 - \frac{4}{\sqrt{{(x - 5)}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$1 - \frac{4}{\sqrt{x - 5}}$

Step 6.2

Set the denominator in $\frac{4}{\sqrt{x - 5}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x - 5} = 0$

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.

${\sqrt{x - 5}}^{2} = 0^{2}$

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$x - 5 = 0^{2}$

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 $0$ .

$x - 5 = 0$

Step 6.3.3

Add $5$ to both sides of the equation.

$x = 5$

Step 6.4

Set the radicand in $\sqrt{x - 5}$ less than $0$ to find where the expression is undefined.

$x - 5 < 0$

Step 6.5

Add $5$ to both sides of the inequality.

$x < 5$

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 $0$ .

$x \leq 5$

$(- \infty, 5]$

$x \leq 5$

$(- \infty, 5]$

Step 7

Critical points to evaluate.

$x = 21, 5$

Step 8

Evaluate the second derivative at $x = 21$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2}{{((21) - 5)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Subtract $5$ from $21$ .

$\frac{2}{16^{\frac{3}{2}}}$

Step 9.1.2

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

$\frac{2}{{(4^{2})}^{\frac{3}{2}}}$

Step 9.1.3

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

$\frac{2}{4^{2 (\frac{3}{2})}}$

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Cancel the common factor.

$\frac{2}{4^{2 (\frac{3}{2})}}$

Step 9.1.4.2

Rewrite the expression.

$\frac{2}{4^{3}}$

Step 9.1.5

Raise $4$ to the power of $3$ .

$\frac{2}{64}$

Step 9.2

Cancel the common factor of $2$ and $64$ .

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Step 9.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{64}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $64$ .

$\frac{2 \cdot 1}{2 \cdot 32}$

Step 9.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 32}$

Step 9.2.2.3

Rewrite the expression.

$\frac{1}{32}$

Step 10

$x = 21$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 21$ is a local minimum

Step 11

Find the y-value when $x = 21$ .

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Step 11.1

Replace the variable $x$ with $21$ in the expression.

$f (21) = (21) - 8 \sqrt{(21) - 5}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Subtract $5$ from $21$ .

$f (21) = 21 - 8 \sqrt{16}$

Step 11.2.1.2

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

$f (21) = 21 - 8 \sqrt{4^{2}}$

Step 11.2.1.3

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

$f (21) = 21 - 8 \cdot 4$

Step 11.2.1.4

Multiply $- 8$ by $4$ .

$f (21) = 21 - 32$

Step 11.2.2

Subtract $32$ from $21$ .

$f (21) = - 11$

Step 11.2.3

The final answer is $- 11$ .

$y = - 11$

Step 12

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.

$\frac{2}{{((5) - 5)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Subtract $5$ from $5$ .

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

Step 13.1.2

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

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

Step 13.1.3

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

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

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

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

Step 13.2.2

Rewrite the expression.

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

Step 13.3

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

$\frac{2}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15