Find the Local Maxima and Minima 4x square root of 1-x^2

$4 x \sqrt{1 - x^{2}}$

Step 1

Write $4 x \sqrt{1 - x^{2}}$ as a function.

$f (x) = 4 x \sqrt{1 - x^{2}}$

Step 2

Find the first derivative of the function.

Differentiate using the Constant Multiple Rule.

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

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

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

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(1 - x^{2})}^{\frac{1}{2}}$ .

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

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

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

$4 (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [1 - x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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

$4 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [1 - x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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

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

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

$4 (x (\frac{1}{2} {(1 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 - x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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

$4 (x (\frac{1}{2} {(1 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Combine the numerators over the common denominator.

$4 (x (\frac{1}{2} {(1 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 - x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Combine fractions.

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

$4 (\frac{x}{2 {(1 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

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

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

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

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

Combine fractions.

Multiply $2$ by $- 1$ .

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

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

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

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

$4 (\frac{- 2 x \cdot x}{2 {(1 - x^{2})}^{\frac{1}{2}}} + {(1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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

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

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

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

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

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

Add $1$ and $1$ .

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

Factor $2$ out of $- 2 x^{2}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

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

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

Subtract $x^{2}$ from $- x^{2}$ .

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

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

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

Simplify.

Apply the distributive property.

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

Simplify each term.

Multiply $- 2$ by $4$ .

$\frac{- 8 x^{2} + 4 \cdot 1}{{(1 - x^{2})}^{\frac{1}{2}}}$

Multiply $4$ by $1$ .

$\frac{- 8 x^{2} + 4}{{(1 - x^{2})}^{\frac{1}{2}}}$

Reorder terms.

$\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = - 8 x^{2} + 4$ and $g (x) = {(- x^{2} + 1)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} (- 8 x^{2} + 4) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{{({(- x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

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

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} (- 8 x^{2} + 4) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} (- 8 x^{2} + 4) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} (- 8 x^{2} + 4) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

Simplify.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} (- 8 x^{2} + 4) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

Differentiate.

By the Sum Rule, the derivative of $- 8 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (\frac{d}{d x} (- 8 x^{2}) + \frac{d}{d x} (4)) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 8 \frac{d}{d x} x^{2} + \frac{d}{d x} (4)) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 8 (2 x) + \frac{d}{d x} (4)) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

Multiply $2$ by $- 8$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x + \frac{d}{d x} (4)) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x + 0) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

Add $- 16 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{d}{d x} {(- x^{2} + 1)}^{\frac{1}{2}}}{- x^{2} + 1}$

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

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{\frac{- 1}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

Combine fractions.

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2} \cdot {(- x^{2} + 1)}^{- \frac{1}{2}} \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{{(- x^{2} + 1)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2} + 1))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (1)))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2} + \frac{d}{d x} (1)))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (- (2 x) + \frac{d}{d x} (1)))}{- x^{2} + 1}$

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (- 2 x + \frac{d}{d x} (1)))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (- 2 x + 0))}{- x^{2} + 1}$

Simplify terms.

Add $- 2 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{1}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot (- 2 x))}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (\frac{- 2}{2 {(- x^{2} + 1)}^{\frac{1}{2}}} \cdot x)}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{- 2 x}{2 {(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Cancel the common factors.

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

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{2 (- x)}{2 ({(- x^{2} + 1)}^{\frac{1}{2}})}}{- x^{2} + 1}$

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{2 (- x)}{2 {(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) \frac{- x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) - (- 8 x^{2} + 4) (- \frac{x}{{(- x^{2} + 1)}^{\frac{1}{2}}})}{- x^{2} + 1}$

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) + 1 (- 8 x^{2} + 4) (\frac{x}{{(- x^{2} + 1)}^{\frac{1}{2}}})}{- x^{2} + 1}$

Multiply $- 8 x^{2} + 4$ by $1$ .

$f'' (x) = \frac{{(- x^{2} + 1)}^{\frac{1}{2}} (- 16 x) + (- 8 x^{2} + 4) (\frac{x}{{(- x^{2} + 1)}^{\frac{1}{2}}})}{- x^{2} + 1}$

Simplify.

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x + (- 8 x^{2} + 4) (\frac{x}{{(- x^{2} + 1)}^{\frac{1}{2}}})}{- x^{2} + 1}$

Multiply $- 8 x^{2} + 4$ by $\frac{x}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x + \frac{(- 8 x^{2} + 4) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Factor $4$ out of $- 8 x^{2} + 4$ .

Factor $4$ out of $- 8 x^{2}$ .

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x + \frac{(4 (- 2 x^{2}) + 4) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Factor $4$ out of $4$ .

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x + \frac{(4 (- 2 x^{2}) + 4 (1)) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Factor $4$ out of $4 (- 2 x^{2}) + 4 (1)$ .

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x + \frac{4 (- 2 x^{2} + 1) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

To write $- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x \cdot \frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}} + \frac{4 (- 2 x^{2} + 1) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Combine $- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x$ and $\frac{{(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}}}{{(- x^{2} + 1)}^{\frac{1}{2}}} + \frac{4 (- 2 x^{2} + 1) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}} + 4 (- 2 x^{2} + 1) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$

Rewrite $\frac{- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}} + 4 (- 2 x^{2} + 1) x}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ in a factored form.

Factor $4 x$ out of $- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}} + 4 (- 2 x^{2} + 1) x$ .

Factor $4 x$ out of $- 16 {(- x^{2} + 1)}^{\frac{1}{2}} x {(- x^{2} + 1)}^{\frac{1}{2}}$ .

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

Factor $4 x$ out of $4 (- 2 x^{2} + 1) x$ .

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

Factor $4 x$ out of $4 x (- 4 {(- x^{2} + 1)}^{\frac{1}{2}} {(- x^{2} + 1)}^{\frac{1}{2}}) + 4 x (- 2 x^{2} + 1)$ .

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

Combine exponents.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $- 4 {(- x^{2} + 1)}^{1}$ .

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 4$ .

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

Multiply $- 4$ by $1$ .

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

Subtract $2 x^{2}$ from $4 x^{2}$ .

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

Add $- 4$ and $1$ .

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

Combine terms.

Rewrite $\frac{\frac{4 x (2 x^{2} - 3)}{{(- x^{2} + 1)}^{\frac{1}{2}}}}{- x^{2} + 1}$ as a product.

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

Multiply $\frac{4 x (2 x^{2} - 3)}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ by $\frac{1}{- x^{2} + 1}$ .

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

Step 4

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

$\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

Differentiate using the Constant Multiple Rule.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Combine fractions.

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

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

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

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

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

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

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

Combine fractions.

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Factor $2$ out of $- 2 x^{2}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

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

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

Subtract $x^{2}$ from $- x^{2}$ .

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

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

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

Simplify.

Apply the distributive property.

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

Simplify each term.

Multiply $- 2$ by $4$ .

$f' (x) = \frac{- 8 x^{2} + 4 \cdot 1}{{(1 - x^{2})}^{\frac{1}{2}}}$

Multiply $4$ by $1$ .

$f' (x) = \frac{- 8 x^{2} + 4}{{(1 - x^{2})}^{\frac{1}{2}}}$

Reorder terms.

$f' (x) = \frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}}$

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}} = 0$ .

Set the first derivative equal to $0$ .

$\frac{- 8 x^{2} + 4}{{(- x^{2} + 1)}^{\frac{1}{2}}} = 0$

Set the numerator equal to zero.

$- 8 x^{2} + 4 = 0$

Solve the equation for $x$ .

Subtract $4$ from both sides of the equation.

$- 8 x^{2} = - 4$

Divide each term in $- 8 x^{2} = - 4$ by $- 8$ and simplify.

Divide each term in $- 8 x^{2} = - 4$ by $- 8$ .

$\frac{- 8 x^{2}}{- 8} = \frac{- 4}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x^{2}}{- 8} = \frac{- 4}{- 8}$

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

$x^{2} = \frac{- 4}{- 8}$

Simplify the right side.

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

Factor $- 4$ out of $- 4$ .

$x^{2} = \frac{- 4 (1)}{- 8}$

Cancel the common factors.

Factor $- 4$ out of $- 8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = \pm \sqrt{\frac{1}{2}}$

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

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

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

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

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

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

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

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

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

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

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

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

Step 7

Find the values where the derivative is undefined.

Convert expressions with fractional exponents to radicals.

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

$\frac{- 8 x^{2} + 4}{\sqrt{{(- x^{2} + 1)}^{1}}}$

Anything raised to $1$ is the base itself.

$\frac{- 8 x^{2} + 4}{\sqrt{- x^{2} + 1}}$

Set the denominator in $\frac{- 8 x^{2} + 4}{\sqrt{- x^{2} + 1}}$ equal to $0$ to find where the expression is undefined.

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

Solve for $x$ .

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$- x^{2} + 1 = 0^{2}$

Simplify the right side.

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

$- x^{2} + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

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

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

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

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

$x = 1$

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

$x = - 1$

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

$x = 1, - 1$

Set the radicand in $\sqrt{- x^{2} + 1}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 1 < 0$

Solve for $x$ .

Subtract $1$ from both sides of the inequality.

$- x^{2} < - 1$

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

Divide each term in $- x^{2} < - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x^{2} > 1$

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

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

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x | > \sqrt{1}$

Simplify the right side.

Any root of $1$ is $1$ .

$| x | > 1$

Write $| x | > 1$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x > 1$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 1$

Write as a piecewise.

${\begin{cases} x > 1 & x \geq 0 \\ - x > 1 & x < 0 \end{cases}$

Find the intersection of $x > 1$ and $x \geq 0$ .

$x > 1$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x < - 1$

Find the union of the solutions.

$x < - 1$ or $x > 1$

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 $0$ .

$x \leq - 1, x \geq 1$

$(- \infty, - 1] \cup [1, \infty)$

$x \leq - 1, x \geq 1$

$(- \infty, - 1] \cup [1, \infty)$

Step 8

Critical points to evaluate.

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

Step 9

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

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

Step 10

Evaluate the second derivative.

Combine $4$ and $\frac{\sqrt{2}}{2}$ .

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

Simplify the denominator.

Simplify each term.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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

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

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

One to any power is one.

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

Simplify the numerator.

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{4 \sqrt{2}}{2}$ by cancelling the common factors.

Factor $2$ out of $4 \sqrt{2}$ .

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

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $2$ .

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

Subtract $3$ from $1$ .

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

Multiply $- 2$ by $2$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $3$ and $1$ .

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$- 4 \cdot 2^{2}$

Factor out negative.

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

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

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

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

$- 2^{2 + 2}$

Add $2$ and $2$ .

$- 2^{4}$

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

$- 1 \cdot 16$

Multiply $- 1$ by $16$ .

$- 16$

Step 11

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

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

Step 12

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

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

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

Simplify the result.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$f (\frac{\sqrt{2}}{2}) = 2 \sqrt{2} \sqrt{1 - \frac{2}{4}}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$f (\frac{\sqrt{2}}{2}) = 2 \sqrt{2} \sqrt{1 - \frac{2 (1)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

Cancel the common factor of $\sqrt{2}$ .

Factor $\sqrt{2}$ out of $2 \sqrt{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The final answer is $2$ .

$y = 2$

Step 13

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

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

Step 14

Evaluate the second derivative.

Simplify the numerator.

Multiply $- 1$ by $4$ .

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

Combine $- 4$ and $\frac{\sqrt{2}}{2}$ .

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

Simplify the denominator.

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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

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

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

One to any power is one.

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

Simplify the numerator.

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 4 \sqrt{2}}{2}$ by cancelling the common factors.

Factor $2$ out of $- 4 \sqrt{2}$ .

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

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Raise $- 1$ to the power of $2$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $2$ .

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

Subtract $3$ from $1$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $4 \sqrt{2} \cdot 2^{\frac{3}{2}}$ .

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

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

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

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

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

$2^{\frac{3}{2} + 2 \cdot \frac{2}{2}} \sqrt{2}$

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

$2^{\frac{3}{2} + \frac{2 \cdot 2}{2}} \sqrt{2}$

Combine the numerators over the common denominator.

$2^{\frac{3 + 2 \cdot 2}{2}} \sqrt{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

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

Add $3$ and $4$ .

$2^{\frac{7}{2}} \sqrt{2}$

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

$2^{\frac{7}{2}} \cdot 2^{\frac{1}{2}}$

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

$2^{\frac{7}{2} + \frac{1}{2}}$

Combine the numerators over the common denominator.

$2^{\frac{7 + 1}{2}}$

Add $7$ and $1$ .

$2^{\frac{8}{2}}$

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

Factor $2$ out of $8$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$2^{4}$

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

$16$

Step 15

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

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

Step 16

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

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

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

Simplify the result.

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$f (- \frac{\sqrt{2}}{2}) = - 2 \sqrt{2} \sqrt{1 - \frac{2}{4}}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$f (- \frac{\sqrt{2}}{2}) = - 2 \sqrt{2} \sqrt{1 - \frac{2 (1)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

Cancel the common factor of $\sqrt{2}$ .

Factor $\sqrt{2}$ out of $- 2 \sqrt{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The final answer is $- 2$ .

$y = - 2$

Step 17

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.

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

Step 18

Evaluate the second derivative.

Simplify each term.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

Reduce the expression by cancelling the common factors.

Add $- 1$ and $1$ .

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

Simplify the expression.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Undefined

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

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

Undefined

Step 19

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

No Local Extrema

Step 20

Find the Local Maxima and Minima 4x square root of 1-x^2

Related Questions

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