Calculus Examples

$x + \sqrt{1 - x}$

Step 1

Write $x + \sqrt{1 - x}$ as a function.

$f (x) = x + \sqrt{1 - x}$

Step 2

Find the first 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 $x + \sqrt{1 - x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{1 - x}]$ .

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

Step 2.1.2

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

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

Step 2.2

Evaluate $\frac{d}{d x} [\sqrt{1 - x}]$ .

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Step 2.2.1

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

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

Step 2.2.2

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 $g (x) = 1 - x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Combine the numerators over the common denominator.

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

Step 2.2.10

Simplify the numerator.

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Step 2.2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.2.10.2

Subtract $2$ from $1$ .

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

Step 2.2.11

Move the negative in front of the fraction.

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

Step 2.2.12

Multiply $- 1$ by $1$ .

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

Step 2.2.13

Subtract $1$ from $0$ .

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

Step 2.2.14

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

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

Step 2.2.15

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

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

Step 2.2.16

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

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

Step 2.2.17

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

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

Step 2.2.18

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

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

Step 2.2.19

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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 $g (x) = 1 - x$ .

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Step 3.2.4.1

To apply the Chain Rule, set $u_{2}$ as $1 - x$ .

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

Step 3.2.4.2

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

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

Step 3.2.4.3

Replace all occurrences of $u_{2}$ with $1 - x$ .

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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Step 3.2.9.1

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

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

Step 3.2.9.2

Cancel the common factor of $2$ .

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Step 3.2.9.2.1

Factor $2$ out of $- 2$ .

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

Step 3.2.9.2.2

Cancel the common factor.

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

Step 3.2.9.2.3

Rewrite the expression.

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

Step 3.2.10

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

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

Step 3.2.11

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

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

Step 3.2.12

Combine the numerators over the common denominator.

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

Step 3.2.13

Simplify the numerator.

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Step 3.2.13.1

Multiply $- 1$ by $2$ .

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

Step 3.2.13.2

Subtract $2$ from $1$ .

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

Step 3.2.14

Move the negative in front of the fraction.

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

Step 3.2.15

Multiply $- 1$ by $1$ .

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

Step 3.2.16

Subtract $1$ from $0$ .

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

Step 3.2.17

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

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

Step 3.2.18

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

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

Step 3.2.19

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

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

Step 3.2.20

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

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

Step 3.2.21

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

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

Step 3.2.22

Move the negative in front of the fraction.

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

Step 3.2.23

Multiply $- 1$ by $- 1$ .

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

Step 3.2.24

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

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

Step 3.2.25

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

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

Step 3.2.26

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

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

Step 3.2.27

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

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Step 3.2.27.1

Move $1 - x$ .

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

Step 3.2.27.2

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

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Step 3.2.27.2.1

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

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

Step 3.2.27.2.2

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

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

Step 3.2.27.3

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

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

Step 3.2.27.4

Combine the numerators over the common denominator.

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

Step 3.2.27.5

Add $2$ and $1$ .

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

Step 3.2.28

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

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

Step 3.2.29

Multiply $2$ by $2$ .

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

Step 3.3

Subtract $\frac{1}{4 {(1 - x)}^{\frac{3}{2}}}$ from $0$ .

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate.

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Step 5.1.1.1

By the Sum Rule, the derivative of $x + \sqrt{1 - x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{1 - x}]$ .

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

Step 5.1.1.2

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

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

Step 5.1.2

Evaluate $\frac{d}{d x} [\sqrt{1 - x}]$ .

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Step 5.1.2.1

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

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

Step 5.1.2.2

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 $g (x) = 1 - x$ .

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Step 5.1.2.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 5.1.2.2.2

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

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

Step 5.1.2.2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 5.1.2.3

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

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

Step 5.1.2.4

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

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

Step 5.1.2.5

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

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

Step 5.1.2.6

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

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

Step 5.1.2.7

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

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

Step 5.1.2.8

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

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

Step 5.1.2.9

Combine the numerators over the common denominator.

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

Step 5.1.2.10

Simplify the numerator.

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Step 5.1.2.10.1

Multiply $- 1$ by $2$ .

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

Step 5.1.2.10.2

Subtract $2$ from $1$ .

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

Step 5.1.2.11

Move the negative in front of the fraction.

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

Step 5.1.2.12

Multiply $- 1$ by $1$ .

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

Step 5.1.2.13

Subtract $1$ from $0$ .

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

Step 5.1.2.14

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

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

Step 5.1.2.15

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

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

Step 5.1.2.16

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

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

Step 5.1.2.17

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

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

Step 5.1.2.18

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

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

Step 5.1.2.19

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Subtract $1$ from both sides of the equation.

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

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

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

Step 6.3.2

The LCM of one and any expression is the expression.

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

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

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Step 6.4.2.1.1

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

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

Step 6.4.2.1.2

Cancel the common factor.

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

Step 6.4.2.1.3

Rewrite the expression.

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

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Multiply $2$ by $- 1$ .

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

Step 6.5

Solve the equation.

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Step 6.5.1

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

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

Step 6.5.2

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

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Step 6.5.2.1

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

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

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor.

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

Step 6.5.2.2.2

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

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

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Dividing two negative values results in a positive value.

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

Step 6.5.3

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

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

Step 6.5.4

Simplify the exponent.

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Step 6.5.4.1

Simplify the left side.

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Step 6.5.4.1.1

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

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Step 6.5.4.1.1.1

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

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Step 6.5.4.1.1.1.1

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

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

Step 6.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 6.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 6.5.4.1.1.2

Simplify.

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

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

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

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Step 6.5.4.2.1.1

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

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

Step 6.5.4.2.1.2

One to any power is one.

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

Step 6.5.4.2.1.3

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

$1 - x = \frac{1}{4}$

Step 6.5.5

Solve for $x$ .

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Step 6.5.5.1

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

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Step 6.5.5.1.1

Subtract $1$ from both sides of the equation.

$- x = \frac{1}{4} - 1$

Step 6.5.5.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- x = \frac{1}{4} - 1 \cdot \frac{4}{4}$

Step 6.5.5.1.3

Combine $- 1$ and $\frac{4}{4}$ .

$- x = \frac{1}{4} + \frac{- 1 \cdot 4}{4}$

Step 6.5.5.1.4

Combine the numerators over the common denominator.

$- x = \frac{1 - 1 \cdot 4}{4}$

Step 6.5.5.1.5

Simplify the numerator.

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Step 6.5.5.1.5.1

Multiply $- 1$ by $4$ .

$- x = \frac{1 - 4}{4}$

Step 6.5.5.1.5.2

Subtract $4$ from $1$ .

$- x = \frac{- 3}{4}$

Step 6.5.5.1.6

Move the negative in front of the fraction.

$- x = - \frac{3}{4}$

Step 6.5.5.2

Divide each term in $- x = - \frac{3}{4}$ by $- 1$ and simplify.

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Step 6.5.5.2.1

Divide each term in $- x = - \frac{3}{4}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- \frac{3}{4}}{- 1}$

Step 6.5.5.2.2

Simplify the left side.

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Step 6.5.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- \frac{3}{4}}{- 1}$

Step 6.5.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{3}{4}}{- 1}$

Step 6.5.5.2.3

Simplify the right side.

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Step 6.5.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{3}{4}}{1}$

Step 6.5.5.2.3.2

Divide $\frac{3}{4}$ by $1$ .

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

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

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

Step 7.1.2

Anything raised to $1$ is the base itself.

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

Step 7.2

Set the denominator in $\frac{1}{2 \sqrt{1 - x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{1 - x} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

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

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

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

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

Step 7.3.2.2.1.2

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

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

Step 7.3.2.2.1.3

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

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Step 7.3.2.2.1.3.1

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

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

Step 7.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.4

Simplify.

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

Step 7.3.2.2.1.5

Apply the distributive property.

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

Step 7.3.2.2.1.6

Multiply.

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Step 7.3.2.2.1.6.1

Multiply $4$ by $1$ .

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

Step 7.3.2.2.1.6.2

Multiply $- 1$ by $4$ .

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

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$4 - 4 x = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Subtract $4$ from both sides of the equation.

$- 4 x = - 4$

Step 7.3.3.2

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

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Step 7.3.3.2.1

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

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

Step 7.3.3.2.2

Simplify the left side.

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Step 7.3.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 7.3.3.2.2.1.1

Cancel the common factor.

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

Step 7.3.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 7.3.3.2.3

Simplify the right side.

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Step 7.3.3.2.3.1

Divide $- 4$ by $- 4$ .

$x = 1$

Step 7.4

Set the radicand in $\sqrt{1 - x}$ less than $0$ to find where the expression is undefined.

$1 - x < 0$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Subtract $1$ from both sides of the inequality.

$- x < - 1$

Step 7.5.2

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

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Step 7.5.2.1

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

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

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 7.5.2.2.2

Divide $x$ by $1$ .

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

Step 7.5.2.3

Simplify the right side.

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Step 7.5.2.3.1

Divide $- 1$ by $- 1$ .

$x > 1$

Step 7.6

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 $0$ .

$x \geq 1$

$[1, \infty)$

$x \geq 1$

$[1, \infty)$

Step 8

Critical points to evaluate.

$x = \frac{3}{4}, 1$

Step 9

Evaluate the second derivative at $x = \frac{3}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1}{4 {(1 - (\frac{3}{4}))}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

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

$- \frac{1}{4 {(\frac{4}{4} - \frac{3}{4})}^{\frac{3}{2}}}$

Step 10.1.2

Combine the numerators over the common denominator.

$- \frac{1}{4 {(\frac{4 - 3}{4})}^{\frac{3}{2}}}$

Step 10.1.3

Subtract $3$ from $4$ .

$- \frac{1}{4 {(\frac{1}{4})}^{\frac{3}{2}}}$

Step 10.1.4

Apply the product rule to $\frac{1}{4}$ .

$- \frac{1}{4 \frac{1^{\frac{3}{2}}}{4^{\frac{3}{2}}}}$

Step 10.1.5

One to any power is one.

$- \frac{1}{4 \frac{1}{4^{\frac{3}{2}}}}$

Step 10.1.6

Simplify the denominator.

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Step 10.1.6.1

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

$- \frac{1}{4 \frac{1}{{(2^{2})}^{\frac{3}{2}}}}$

Step 10.1.6.2

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

$- \frac{1}{4 \frac{1}{2^{2 (\frac{3}{2})}}}$

Step 10.1.6.3

Cancel the common factor of $2$ .

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Step 10.1.6.3.1

Cancel the common factor.

$- \frac{1}{4 \frac{1}{2^{2 (\frac{3}{2})}}}$

Step 10.1.6.3.2

Rewrite the expression.

$- \frac{1}{4 \frac{1}{2^{3}}}$

Step 10.1.6.4

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

$- \frac{1}{4 (\frac{1}{8})}$

Step 10.2

Simplify terms.

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Step 10.2.1

Combine $4$ and $\frac{1}{8}$ .

$- \frac{1}{\frac{4}{8}}$

Step 10.2.2

Cancel the common factor of $4$ and $8$ .

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Step 10.2.2.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{8}}$

Step 10.2.2.2

Cancel the common factors.

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Step 10.2.2.2.1

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 10.2.2.2.2

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 10.2.2.2.3

Rewrite the expression.

$- \frac{1}{\frac{1}{2}}$

Step 10.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 10.4

Multiply $- (1 \cdot 2)$ .

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Step 10.4.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 10.4.2

Multiply $- 1$ by $2$ .

$- 2$

Step 11

$x = \frac{3}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3}{4}$ is a local maximum

Step 12

Find the y-value when $x = \frac{3}{4}$ .

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Step 12.1

Replace the variable $x$ with $\frac{3}{4}$ in the expression.

$f (\frac{3}{4}) = (\frac{3}{4}) + \sqrt{1 - (\frac{3}{4})}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

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

$f (\frac{3}{4}) = \frac{3}{4} + \sqrt{\frac{4}{4} - \frac{3}{4}}$

Step 12.2.1.2

Combine the numerators over the common denominator.

$f (\frac{3}{4}) = \frac{3}{4} + \sqrt{\frac{4 - 3}{4}}$

Step 12.2.1.3

Subtract $3$ from $4$ .

$f (\frac{3}{4}) = \frac{3}{4} + \sqrt{\frac{1}{4}}$

Step 12.2.1.4

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$f (\frac{3}{4}) = \frac{3}{4} + \frac{\sqrt{1}}{\sqrt{4}}$

Step 12.2.1.5

Any root of $1$ is $1$ .

$f (\frac{3}{4}) = \frac{3}{4} + \frac{1}{\sqrt{4}}$

Step 12.2.1.6

Simplify the denominator.

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Step 12.2.1.6.1

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

$f (\frac{3}{4}) = \frac{3}{4} + \frac{1}{\sqrt{2^{2}}}$

Step 12.2.1.6.2

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

$f (\frac{3}{4}) = \frac{3}{4} + \frac{1}{2}$

Step 12.2.2

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

$f (\frac{3}{4}) = \frac{3}{4} + \frac{1}{2} \cdot \frac{2}{2}$

Step 12.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.3.1

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

$f (\frac{3}{4}) = \frac{3}{4} + \frac{2}{2 \cdot 2}$

Step 12.2.3.2

Multiply $2$ by $2$ .

$f (\frac{3}{4}) = \frac{3}{4} + \frac{2}{4}$

Step 12.2.4

Combine the numerators over the common denominator.

$f (\frac{3}{4}) = \frac{3 + 2}{4}$

Step 12.2.5

Add $3$ and $2$ .

$f (\frac{3}{4}) = \frac{5}{4}$

Step 12.2.6

The final answer is $\frac{5}{4}$ .

$y = \frac{5}{4}$

Step 13

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1}{4 {(1 - (1))}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify the expression.

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Step 14.1.1

Multiply $- 1$ by $1$ .

$- \frac{1}{4 {(1 - 1)}^{\frac{3}{2}}}$

Step 14.1.2

Subtract $1$ from $1$ .

$- \frac{1}{4 \cdot 0^{\frac{3}{2}}}$

Step 14.1.3

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

$- \frac{1}{4 \cdot {(0^{2})}^{\frac{3}{2}}}$

Step 14.1.4

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

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 14.2

Cancel the common factor of $2$ .

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Step 14.2.1

Cancel the common factor.

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Step 14.2.2

Rewrite the expression.

$- \frac{1}{4 \cdot 0^{3}}$

Step 14.3

Simplify the expression.

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Step 14.3.1

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

$- \frac{1}{4 \cdot 0}$

Step 14.3.2

Multiply $4$ by $0$ .

$- \frac{1}{0}$

Step 14.3.3

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

Undefined

$- \frac{1}{0}$

Step 14.4

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

Undefined

Step 15

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

No Local Extrema

Step 16