Calculus Examples

$f (x) = \arcsin (x) - 2 x$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 2$ by $1$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

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

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Step 2.2.3.1

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

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

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 $n = - 1$ .

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

Step 2.2.3.3

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

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

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

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Step 2.2.4.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 2.2.4.3

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

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

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Step 2.2.9.1

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

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

Step 2.2.9.2

Cancel the common factor of $2$ .

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Step 2.2.9.2.1

Factor $2$ out of $- 2$ .

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

Step 2.2.9.2.2

Cancel the common factor.

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

Step 2.2.9.2.3

Rewrite the expression.

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

Step 2.2.10

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

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

Step 2.2.11

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

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

Step 2.2.12

Combine the numerators over the common denominator.

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

Step 2.2.13

Simplify the numerator.

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Step 2.2.13.1

Multiply $- 1$ by $2$ .

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

Step 2.2.13.2

Subtract $2$ from $1$ .

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

Step 2.2.14

Move the negative in front of the fraction.

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

Step 2.2.15

Multiply $2$ by $- 1$ .

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

Step 2.2.16

Subtract $2 x$ from $0$ .

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

Step 2.2.17

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

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

Step 2.2.18

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

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

Step 2.2.19

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

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

Step 2.2.20

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

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

Step 2.2.21

Factor $2$ out of $- 2 x$ .

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

Step 2.2.22

Cancel the common factors.

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Step 2.2.22.1

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

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

Step 2.2.22.2

Cancel the common factor.

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

Step 2.2.22.3

Rewrite the expression.

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

Step 2.2.23

Move the negative in front of the fraction.

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

Step 2.2.24

Multiply $- 1$ by $- 1$ .

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

Step 2.2.25

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

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

Step 2.2.26

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

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

Step 2.2.27

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

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

Step 2.2.28

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

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Step 2.2.28.1

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

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Step 2.2.28.1.1

Raise $1 - x^{2}$ to the power of $1$ .

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

Step 2.2.28.1.2

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

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

Step 2.2.28.2

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

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

Step 2.2.28.3

Combine the numerators over the common denominator.

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

Step 2.2.28.4

Add $1$ and $2$ .

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

Step 2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Add $\frac{x}{{(1 - x^{2})}^{\frac{3}{2}}}$ and $0$ .

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

Step 2.4.2

Reorder terms.

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

Step 3

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

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

Step 4

Solve for $\sqrt{1 - x^{2}}$ .

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Step 4.1

Add $2$ to both sides of the equation.

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

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

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

$\sqrt{1 - x^{2}}, 1$

Step 4.2.2

The LCM of one and any expression is the expression.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Rewrite the expression.

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

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $2 \sqrt{1 - x^{2}} = 1$ .

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

Step 4.4.2

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

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Step 4.4.2.1

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

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

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.2.1.2

Divide $\sqrt{1 - x^{2}}$ by $1$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.2.1

Cancel the common factor.

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

Step 6.2.1.1.2.2

Rewrite the expression.

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

Step 6.2.1.2

Simplify.

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

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

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

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

Step 6.3.1.2

One to any power is one.

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

Step 6.3.1.3

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

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

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $1$ from both sides of the equation.

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

Combine the numerators over the common denominator.

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

Step 7.1.5

Simplify the numerator.

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Step 7.1.5.1

Multiply $- 1$ by $4$ .

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

Step 7.1.5.2

Subtract $4$ from $1$ .

$- x^{2} = \frac{- 3}{4}$

Step 7.1.6

Move the negative in front of the fraction.

$- x^{2} = - \frac{3}{4}$

Step 7.2

Divide each term in $- x^{2} = - \frac{3}{4}$ by $- 1$ and simplify.

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Step 7.2.1

Divide each term in $- x^{2} = - \frac{3}{4}$ by $- 1$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.2.2

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

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{\frac{3}{4}}{1}$

Step 7.2.3.2

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

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

Step 7.3

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

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

Step 7.4

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 7.4.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

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

Step 7.4.2

Simplify the denominator.

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Step 7.4.2.1

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

$x = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 7.4.2.2

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

$x = \pm \frac{\sqrt{3}}{2}$

Step 7.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{3}}{2}$

Step 7.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{3}}{2}$

Step 7.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Step 8

Evaluate the second derivative at $x = \frac{\sqrt{3}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{2} \cdot \frac{1}{{(- {(\frac{\sqrt{3}}{2})}^{2} + 1)}^{\frac{3}{2}}}$

Step 9.2

Combine fractions.

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Step 9.2.1

Combine.

$\frac{\sqrt{3} \cdot 1}{2 {(- {(\frac{\sqrt{3}}{2})}^{2} + 1)}^{\frac{3}{2}}}$

Step 9.2.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 9.3

Simplify the denominator.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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

Step 9.3.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 9.3.1.2.2

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

$\frac{\sqrt{3}}{2 {(- \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 9.3.1.2.3

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

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

Step 9.3.1.2.4

Cancel the common factor of $2$ .

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Step 9.3.1.2.4.1

Cancel the common factor.

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

Step 9.3.1.2.4.2

Rewrite the expression.

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

Step 9.3.1.2.5

Evaluate the exponent.

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

Step 9.3.1.3

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

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

Step 9.3.2

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

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

Step 9.3.3

Combine the numerators over the common denominator.

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

Step 9.3.4

Add $- 3$ and $4$ .

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

Step 9.3.5

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

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

Step 9.3.6

One to any power is one.

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

Step 9.3.7

Simplify the denominator.

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Step 9.3.7.1

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

$\frac{\sqrt{3}}{2 \frac{1}{{(2^{2})}^{\frac{3}{2}}}}$

Step 9.3.7.2

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

$\frac{\sqrt{3}}{2 \frac{1}{2^{2 (\frac{3}{2})}}}$

Step 9.3.7.3

Cancel the common factor of $2$ .

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Step 9.3.7.3.1

Cancel the common factor.

$\frac{\sqrt{3}}{2 \frac{1}{2^{2 (\frac{3}{2})}}}$

Step 9.3.7.3.2

Rewrite the expression.

$\frac{\sqrt{3}}{2 \frac{1}{2^{3}}}$

Step 9.3.7.4

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

$\frac{\sqrt{3}}{2 (\frac{1}{8})}$

Step 9.4

Simplify terms.

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Step 9.4.1

Combine $2$ and $\frac{1}{8}$ .

$\frac{\sqrt{3}}{\frac{2}{8}}$

Step 9.4.2

Cancel the common factor of $2$ and $8$ .

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Step 9.4.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt{3}}{\frac{2 (1)}{8}}$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $2$ out of $8$ .

$\frac{\sqrt{3}}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 9.4.2.2.2

Cancel the common factor.

$\frac{\sqrt{3}}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 9.4.2.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{\frac{1}{4}}$

Step 9.5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{3} \cdot 4$

Step 9.6

Move $4$ to the left of $\sqrt{3}$ .

$4 \sqrt{3}$

Step 10

$x = \frac{\sqrt{3}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{\sqrt{3}}{2}$ is a local minimum

Step 11

Find the y-value when $x = \frac{\sqrt{3}}{2}$ .

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Step 11.1

Replace the variable $x$ with $\frac{\sqrt{3}}{2}$ in the expression.

$f (\frac{\sqrt{3}}{2}) = \arcsin (\frac{\sqrt{3}}{2}) - 2 \frac{\sqrt{3}}{2}$

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

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$f (\frac{\sqrt{3}}{2}) = \frac{π}{3} - 2 \frac{\sqrt{3}}{2}$

Step 11.2.1.2

Cancel the common factor of $2$ .

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Step 11.2.1.2.1

Factor $2$ out of $- 2$ .

$f (\frac{\sqrt{3}}{2}) = \frac{π}{3} + 2 (- 1) (\frac{\sqrt{3}}{2})$

Step 11.2.1.2.2

Cancel the common factor.

$f (\frac{\sqrt{3}}{2}) = \frac{π}{3} + 2 \cdot (- 1 \frac{\sqrt{3}}{2})$

Step 11.2.1.2.3

Rewrite the expression.

$f (\frac{\sqrt{3}}{2}) = \frac{π}{3} - 1 \sqrt{3}$

Step 11.2.1.3

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$f (\frac{\sqrt{3}}{2}) = \frac{π}{3} - \sqrt{3}$

Step 11.2.2

The final answer is $\frac{π}{3} - \sqrt{3}$ .

$y = \frac{π}{3} - \sqrt{3}$

Step 12

Evaluate the second derivative at $x = - \frac{\sqrt{3}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- {(- \frac{\sqrt{3}}{2})}^{2} + 1)}^{\frac{3}{2}}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 13.2.1.1.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- ({(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}) + 1)}^{\frac{3}{2}}}$

Step 13.2.1.1.2

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- ({(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}) + 1)}^{\frac{3}{2}}}$

Step 13.2.1.2

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

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Step 13.2.1.2.1

Move ${(- 1)}^{2}$ .

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{({(- 1)}^{2} \cdot - 1 \frac{{\sqrt{3}}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.2.2

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

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Step 13.2.1.2.2.1

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{({(- 1)}^{2} \cdot {(- 1)}^{1} \frac{{\sqrt{3}}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.2.2.2

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{({(- 1)}^{2 + 1} \frac{{\sqrt{3}}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.2.3

Add $2$ and $1$ .

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{({(- 1)}^{3} \frac{{\sqrt{3}}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.3

Raise $- 1$ to the power of $3$ .

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{{\sqrt{3}}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 13.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4.2

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4.3

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{3^{\frac{2}{2}}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4.4

Cancel the common factor of $2$ .

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Step 13.2.1.4.4.1

Cancel the common factor.

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{3^{\frac{2}{2}}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4.4.2

Rewrite the expression.

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{3^{1}}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.4.5

Evaluate the exponent.

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{{(- \frac{3}{2^{2}} + 1)}^{\frac{3}{2}}}$

Step 13.2.1.5

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

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

Step 13.2.2

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

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

Step 13.2.3

Combine the numerators over the common denominator.

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

Step 13.2.4

Add $- 3$ and $4$ .

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

Step 13.2.5

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

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

Step 13.2.6

One to any power is one.

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

Step 13.2.7

Simplify the denominator.

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Step 13.2.7.1

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

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

Step 13.2.7.2

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

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

Step 13.2.7.3

Cancel the common factor of $2$ .

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Step 13.2.7.3.1

Cancel the common factor.

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

Step 13.2.7.3.2

Rewrite the expression.

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

Step 13.2.7.4

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

$- \frac{\sqrt{3}}{2} \cdot \frac{1}{\frac{1}{8}}$

Step 13.3

Reduce the expression by cancelling the common factors.

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Step 13.3.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sqrt{3}}{2} (1 \cdot 8)$

Step 13.3.2

Multiply $8$ by $1$ .

$- \frac{\sqrt{3}}{2} \cdot 8$

Step 13.3.3

Cancel the common factor of $2$ .

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Step 13.3.3.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$\frac{- \sqrt{3}}{2} \cdot 8$

Step 13.3.3.2

Factor $2$ out of $8$ .

$\frac{- \sqrt{3}}{2} \cdot (2 (4))$

Step 13.3.3.3

Cancel the common factor.

$\frac{- \sqrt{3}}{2} \cdot (2 \cdot 4)$

Step 13.3.3.4

Rewrite the expression.

$- \sqrt{3} \cdot 4$

Step 13.3.4

Multiply $4$ by $- 1$ .

$- 4 \sqrt{3}$

Step 14

$x = - \frac{\sqrt{3}}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{\sqrt{3}}{2}$ is a local maximum

Step 15

Find the y-value when $x = - \frac{\sqrt{3}}{2}$ .

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Step 15.1

Replace the variable $x$ with $- \frac{\sqrt{3}}{2}$ in the expression.

$f (- \frac{\sqrt{3}}{2}) = \arcsin (- \frac{\sqrt{3}}{2}) - 2 (- \frac{\sqrt{3}}{2})$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

The exact value of $\arcsin (- \frac{\sqrt{3}}{2})$ is $- \frac{π}{3}$ .

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} - 2 (- \frac{\sqrt{3}}{2})$

Step 15.2.1.2

Cancel the common factor of $2$ .

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Step 15.2.1.2.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} - 2 \frac{- \sqrt{3}}{2}$

Step 15.2.1.2.2

Factor $2$ out of $- 2$ .

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} + 2 (- 1) (\frac{- \sqrt{3}}{2})$

Step 15.2.1.2.3

Cancel the common factor.

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} + 2 \cdot (- 1 \frac{- \sqrt{3}}{2})$

Step 15.2.1.2.4

Rewrite the expression.

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} - 1 (- \sqrt{3})$

Step 15.2.1.3

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} + 1 \sqrt{3}$

Step 15.2.1.4

Multiply $\sqrt{3}$ by $1$ .

$f (- \frac{\sqrt{3}}{2}) = - \frac{π}{3} + \sqrt{3}$

Step 15.2.2

The final answer is $- \frac{π}{3} + \sqrt{3}$ .

$y = - \frac{π}{3} + \sqrt{3}$

Step 16

These are the local extrema for $f (x) = \arcsin (x) - 2 x$ .

$(\frac{\sqrt{3}}{2}, \frac{π}{3} - \sqrt{3})$ is a local minima

$(- \frac{\sqrt{3}}{2}, - \frac{π}{3} + \sqrt{3})$ is a local maxima

Step 17