Find the Local Maxima and Minima x square root of 64-x^2

$x \sqrt{64 - x^{2}}$

Step 1

Write $x \sqrt{64 - x^{2}}$ as a function.

$f (x) = x \sqrt{64 - x^{2}}$

Step 2

Find the first derivative of the function.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 - x^{2}}$ as ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}}]$

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 $g (x) = {(64 - x^{2})}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(64 - x^{2})}^{\frac{1}{2}}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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 $g (x) = 64 - x^{2}$ .

To apply the Chain Rule, set $u$ as $64 - x^{2}$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Replace all occurrences of $u$ with $64 - x^{2}$ .

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(64 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine fractions.

Move the negative in front of the fraction.

$x (\frac{1}{2} {(64 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine $\frac{1}{2}$ and ${(64 - x^{2})}^{- \frac{1}{2}}$ .

$x (\frac{{(64 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Move ${(64 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine $\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

By the Sum Rule, the derivative of $64 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]$ .

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- (2 x)) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine fractions.

Multiply $2$ by $- 1$ .

$\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- 2 x) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine $- 2$ and $\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}} x + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Combine $\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{- 2 x \cdot x}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{- 2 (x^{1} x)}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{- 2 (x^{1} x^{1})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{- 2 x^{1 + 1}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Add $1$ and $1$ .

$\frac{- 2 x^{2}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Cancel the common factors.

Factor $2$ out of $2 {(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- x^{2})}{2 ({(64 - x^{2})}^{\frac{1}{2}})} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Cancel the common factor.

$\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Rewrite the expression.

$\frac{- x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Move the negative in front of the fraction.

$- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x]$

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

$- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \cdot 1$

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}}$

To write ${(64 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{{(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by ${(64 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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

$\frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{- x^{2} + {(64 - x^{2})}^{\frac{1 + 1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Add $1$ and $1$ .

$\frac{- x^{2} + {(64 - x^{2})}^{\frac{2}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Divide $2$ by $2$ .

$\frac{- x^{2} + {(64 - x^{2})}^{1}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Simplify ${(64 - x^{2})}^{1}$ .

$\frac{- x^{2} + 64 - x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}}$

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

$\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}}$

Reorder terms.

$\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}}$

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 $f (x) = - 2 x^{2} + 64$ and $g (x) = {(- x^{2} + 64)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 64) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{{({(- x^{2} + 64)}^{\frac{1}{2}})}^{2}}$

Multiply the exponents in ${({(- x^{2} + 64)}^{\frac{1}{2}})}^{2}$ .

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 64) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 64) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2} \cdot 2}}$

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 64) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

Simplify.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} \frac{d}{d x} (- 2 x^{2} + 64) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

Differentiate.

By the Sum Rule, the derivative of $- 2 x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [64]$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (\frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (64)) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 2 \frac{d}{d x} x^{2} + \frac{d}{d x} (64)) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 2 (2 x) + \frac{d}{d x} (64)) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x + \frac{d}{d x} (64)) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x + 0) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

Add $- 4 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{d}{d x} {(- x^{2} + 64)}^{\frac{1}{2}}}{- x^{2} + 64}$

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 $g (x) = - x^{2} + 64$ .

To apply the Chain Rule, set $u$ as $- x^{2} + 64$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Replace all occurrences of $u$ with $- x^{2} + 64$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{1}{2} - 1} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{1 - 2}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{\frac{- 1}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Combine fractions.

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2} \cdot {(- x^{2} + 64)}^{- \frac{1}{2}} \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Combine $\frac{1}{2}$ and ${(- x^{2} + 64)}^{- \frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{{(- x^{2} + 64)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

Move ${(- x^{2} + 64)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2} + 64))}{- x^{2} + 64}$

By the Sum Rule, the derivative of $- x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [64]$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (- x^{2}) + \frac{d}{d x} (64)))}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2} + \frac{d}{d x} (64)))}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (- (2 x) + \frac{d}{d x} (64)))}{- x^{2} + 64}$

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (- 2 x + \frac{d}{d x} (64)))}{- x^{2} + 64}$

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (- 2 x + 0))}{- x^{2} + 64}$

Simplify terms.

Add $- 2 x$ and $0$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot (- 2 x))}{- x^{2} + 64}$

Combine $- 2$ and $\frac{1}{2 {(- x^{2} + 64)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (\frac{- 2}{2 {(- x^{2} + 64)}^{\frac{1}{2}}} \cdot x)}{- x^{2} + 64}$

Combine $\frac{- 2}{2 {(- x^{2} + 64)}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{- 2 x}{2 {(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

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

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{2 (- x)}{2 {(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Cancel the common factors.

Factor $2$ out of $2 {(- x^{2} + 64)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{2 (- x)}{2 ({(- x^{2} + 64)}^{\frac{1}{2}})}}{- x^{2} + 64}$

Cancel the common factor.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{2 (- x)}{2 {(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Rewrite the expression.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) \frac{- x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Move the negative in front of the fraction.

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) - (- 2 x^{2} + 64) (- \frac{x}{{(- x^{2} + 64)}^{\frac{1}{2}}})}{- x^{2} + 64}$

Multiply $- 1$ by $- 1$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) + 1 (- 2 x^{2} + 64) (\frac{x}{{(- x^{2} + 64)}^{\frac{1}{2}}})}{- x^{2} + 64}$

Multiply $- 2 x^{2} + 64$ by $1$ .

$f'' (x) = \frac{{(- x^{2} + 64)}^{\frac{1}{2}} (- 4 x) + (- 2 x^{2} + 64) (\frac{x}{{(- x^{2} + 64)}^{\frac{1}{2}}})}{- x^{2} + 64}$

Simplify.

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x + (- 2 x^{2} + 64) (\frac{x}{{(- x^{2} + 64)}^{\frac{1}{2}}})}{- x^{2} + 64}$

Multiply $- 2 x^{2} + 64$ by $\frac{x}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x + \frac{(- 2 x^{2} + 64) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Factor $2$ out of $- 2 x^{2} + 64$ .

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

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 64) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Factor $2$ out of $64$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x + \frac{(2 (- x^{2}) + 2 (32)) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Factor $2$ out of $2 (- x^{2}) + 2 (32)$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x + \frac{2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

To write $- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x \cdot \frac{{(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Combine $- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x$ and $\frac{{(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x {(- x^{2} + 64)}^{\frac{1}{2}}}{{(- x^{2} + 64)}^{\frac{1}{2}}} + \frac{2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x {(- x^{2} + 64)}^{\frac{1}{2}} + 2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Rewrite $\frac{- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x {(- x^{2} + 64)}^{\frac{1}{2}} + 2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ in a factored form.

Factor $2 x$ out of $- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x {(- x^{2} + 64)}^{\frac{1}{2}} + 2 (- x^{2} + 32) x$ .

Factor $2 x$ out of $- 4 {(- x^{2} + 64)}^{\frac{1}{2}} x {(- x^{2} + 64)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{1}{2}} {(- x^{2} + 64)}^{\frac{1}{2}}) + 2 (- x^{2} + 32) x}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Factor $2 x$ out of $2 (- x^{2} + 32) x$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{1}{2}} {(- x^{2} + 64)}^{\frac{1}{2}}) + 2 x (- x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Factor $2 x$ out of $2 x (- 2 {(- x^{2} + 64)}^{\frac{1}{2}} {(- x^{2} + 64)}^{\frac{1}{2}}) + 2 x (- x^{2} + 32)$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{1}{2}} {(- x^{2} + 64)}^{\frac{1}{2}} - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Combine exponents.

Multiply ${(- x^{2} + 64)}^{\frac{1}{2}}$ by ${(- x^{2} + 64)}^{\frac{1}{2}}$ by adding the exponents.

Move ${(- x^{2} + 64)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{2 x (- 2 ({(- x^{2} + 64)}^{\frac{1}{2}} {(- x^{2} + 64)}^{\frac{1}{2}}) - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

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

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{1}{2} + \frac{1}{2}} - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{1 + 1}{2}} - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{2 x (- 2 {(- x^{2} + 64)}^{\frac{2}{2}} - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 64) - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Simplify $- 2 {(- x^{2} + 64)}^{1}$ .

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2} + 64) - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Simplify the numerator.

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x (- 2 (- x^{2}) - 2 \cdot 64 - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 2 \cdot 64 - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Multiply $- 2$ by $64$ .

$f'' (x) = \frac{\frac{2 x (2 x^{2} - 128 - x^{2} + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

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

$f'' (x) = \frac{\frac{2 x (x^{2} - 128 + 32)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Add $- 128$ and $32$ .

$f'' (x) = \frac{\frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$

Combine terms.

Rewrite $\frac{\frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}}}}{- x^{2} + 64}$ as a product.

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}}} \cdot \frac{1}{- x^{2} + 64}$

Multiply $\frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ by $\frac{1}{- x^{2} + 64}$ .

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}} (- x^{2} + 64)}$

Multiply ${(- x^{2} + 64)}^{\frac{1}{2}}$ by $- x^{2} + 64$ by adding the exponents.

Multiply ${(- x^{2} + 64)}^{\frac{1}{2}}$ by $- x^{2} + 64$ .

Raise $- x^{2} + 64$ to the power of $1$ .

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2}} (- x^{2} + 64)}$

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

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2} + 1}}$

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

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{1 + 2}{2}}}$

Add $1$ and $2$ .

$f'' (x) = \frac{2 x (x^{2} - 96)}{{(- x^{2} + 64)}^{\frac{3}{2}}}$

Step 4

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

$\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 - x^{2}}$ as ${(64 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x {(64 - x^{2})}^{\frac{1}{2}})$

$f' (x) = x \frac{d}{d x} ({(64 - x^{2})}^{\frac{1}{2}}) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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 $g (x) = 64 - x^{2}$ .

To apply the Chain Rule, set $u$ as $64 - x^{2}$ .

$f' (x) = x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Replace all occurrences of $u$ with $64 - x^{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine the numerators over the common denominator.

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Subtract $2$ from $1$ .

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine fractions.

Move the negative in front of the fraction.

$f' (x) = x (\frac{1}{2} \cdot {(64 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine $\frac{1}{2}$ and ${(64 - x^{2})}^{- \frac{1}{2}}$ .

$f' (x) = x (\frac{{(64 - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Move ${(64 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x (\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (64 - x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine $\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (64 - x^{2}) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

By the Sum Rule, the derivative of $64 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (64) + \frac{d}{d x} (- x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{2})) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2}) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2}) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot (- (2 x)) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine fractions.

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine $- 2$ and $\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \cdot x + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Combine $\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{- 2 x \cdot x}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{- 2 (x \cdot x)}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{- 2 x^{1 + 1}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Add $1$ and $1$ .

$f' (x) = \frac{- 2 x^{2}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = \frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Cancel the common factors.

Factor $2$ out of $2 {(64 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{2 (- x^{2})}{2 ({(64 - x^{2})}^{\frac{1}{2}})} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Cancel the common factor.

$f' (x) = \frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Rewrite the expression.

$f' (x) = \frac{- x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x)$

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

$f' (x) = - \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \cdot 1$

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$f' (x) = - \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}}$

To write ${(64 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{{(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by ${(64 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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

$f' (x) = \frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$f' (x) = \frac{- x^{2} + {(64 - x^{2})}^{\frac{1 + 1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Add $1$ and $1$ .

$f' (x) = \frac{- x^{2} + {(64 - x^{2})}^{\frac{2}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Divide $2$ by $2$ .

$f' (x) = \frac{- x^{2} + 64 - x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}}$

Simplify ${(64 - x^{2})}^{1}$ .

$f' (x) = \frac{- x^{2} + 64 - x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}}$

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

$f' (x) = \frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}}$

Reorder terms.

$f' (x) = \frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}}$

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}}$ .

$\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}} = 0$ .

Set the first derivative equal to $0$ .

$\frac{- 2 x^{2} + 64}{{(- x^{2} + 64)}^{\frac{1}{2}}} = 0$

Set the numerator equal to zero.

$- 2 x^{2} + 64 = 0$

Solve the equation for $x$ .

Subtract $64$ from both sides of the equation.

$- 2 x^{2} = - 64$

Divide each term in $- 2 x^{2} = - 64$ by $- 2$ and simplify.

Divide each term in $- 2 x^{2} = - 64$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 64}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 64}{- 2}$

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

$x^{2} = \frac{- 64}{- 2}$

Simplify the right side.

Divide $- 64$ by $- 2$ .

$x^{2} = 32$

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

$x = \pm \sqrt{32}$

Simplify $\pm \sqrt{32}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \pm \sqrt{16 (2)}$

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

$x = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$x = \pm 4 \sqrt{2}$

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.

$x = 4 \sqrt{2}$

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

$x = - 4 \sqrt{2}$

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

$x = 4 \sqrt{2}, - 4 \sqrt{2}$

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.

$\frac{- 2 x^{2} + 64}{\sqrt{{(- x^{2} + 64)}^{1}}}$

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 64}{\sqrt{- x^{2} + 64}}$

Set the denominator in $\frac{- 2 x^{2} + 64}{\sqrt{- x^{2} + 64}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 64} = 0$

Solve for $x$ .

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

${\sqrt{- x^{2} + 64}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x^{2} + 64}$ as ${(- x^{2} + 64)}^{\frac{1}{2}}$ .

${({(- x^{2} + 64)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(- x^{2} + 64)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- x^{2} + 64)}^{\frac{1}{2}})}^{2}$ .

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

${(- x^{2} + 64)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- x^{2} + 64)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(- x^{2} + 64)}^{1} = 0^{2}$

Simplify.

$- x^{2} + 64 = 0^{2}$

Simplify the right side.

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

$- x^{2} + 64 = 0$

Solve for $x$ .

Subtract $64$ from both sides of the equation.

$- x^{2} = - 64$

Divide each term in $- x^{2} = - 64$ by $- 1$ and simplify.

Divide each term in $- x^{2} = - 64$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 64}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 64}{- 1}$

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

$x^{2} = \frac{- 64}{- 1}$

Simplify the right side.

Divide $- 64$ by $- 1$ .

$x^{2} = 64$

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

$x = \pm \sqrt{64}$

Simplify $\pm \sqrt{64}$ .

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

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

$x = \pm 8$

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.

$x = 8$

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

$x = - 8$

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

$x = 8, - 8$

Set the radicand in $\sqrt{- x^{2} + 64}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 64 < 0$

Solve for $x$ .

Subtract $64$ from both sides of the inequality.

$- x^{2} < - 64$

Divide each term in $- x^{2} < - 64$ by $- 1$ and simplify.

Divide each term in $- x^{2} < - 64$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 64}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 64}{- 1}$

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

$x^{2} > \frac{- 64}{- 1}$

Simplify the right side.

Divide $- 64$ by $- 1$ .

$x^{2} > 64$

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

$\sqrt{x^{2}} > \sqrt{64}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x | > \sqrt{64}$

Simplify the right side.

Simplify $\sqrt{64}$ .

Rewrite $64$ as $8^{2}$ .

$| x | > \sqrt{8^{2}}$

Pull terms out from under the radical.

$| x | > | 8 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$| x | > 8$

Write $| x | > 8$ 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.

$x > 8$

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 $- 1$ .

$- x > 8$

Write as a piecewise.

${\begin{cases} x > 8 & x \geq 0 \\ - x > 8 & x < 0 \end{cases}$

Find the intersection of $x > 8$ and $x \geq 0$ .

$x > 8$

Divide each term in $- x > 8$ by $- 1$ and simplify.

Divide each term in $- x > 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{8}{- 1}$

Divide $x$ by $1$ .

$x < \frac{8}{- 1}$

Simplify the right side.

Divide $8$ by $- 1$ .

$x < - 8$

Find the union of the solutions.

$x < - 8$ or $x > 8$

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 $0$ .

$x \leq - 8, x \geq 8$

$(- \infty, - 8] \cup [8, \infty)$

$x \leq - 8, x \geq 8$

$(- \infty, - 8] \cup [8, \infty)$

Step 8

Critical points to evaluate.

$x = 4 \sqrt{2}, - 4 \sqrt{2}, - 8, 8$

Step 9

Evaluate the second derivative at $x = 4 \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (4 \sqrt{2}) ({(4 \sqrt{2})}^{2} - 96)}{{(- {(4 \sqrt{2})}^{2} + 64)}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

Multiply $2$ by $4$ .

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- {(4 \sqrt{2})}^{2} + 64)}^{\frac{3}{2}}}$

Simplify the denominator.

Simplify each term.

Apply the product rule to $4 \sqrt{2}$ .

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (4^{2} {\sqrt{2}}^{2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 {\sqrt{2}}^{2}) + 64)}^{\frac{3}{2}}}$

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

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

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 {(2^{\frac{1}{2}})}^{2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{1}{2} \cdot 2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{2}{2}}) + 64)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{2}{2}}) + 64)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{1}) + 64)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2) + 64)}^{\frac{3}{2}}}$

Multiply $- (16 \cdot 2)$ .

Multiply $16$ by $2$ .

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- 1 \cdot 32 + 64)}^{\frac{3}{2}}}$

Multiply $- 1$ by $32$ .

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{{(- 32 + 64)}^{\frac{3}{2}}}$

Add $- 32$ and $64$ .

$\frac{8 \sqrt{2} ({(4 \sqrt{2})}^{2} - 96)}{32^{\frac{3}{2}}}$

Simplify the numerator.

Apply the product rule to $4 \sqrt{2}$ .

$\frac{8 \sqrt{2} (4^{2} {\sqrt{2}}^{2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} (16 {\sqrt{2}}^{2} - 96)}{32^{\frac{3}{2}}}$

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

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

$\frac{8 \sqrt{2} (16 {(2^{\frac{1}{2}})}^{2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} (16 \cdot 2^{\frac{1}{2} \cdot 2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{8 \sqrt{2} (16 \cdot 2^{\frac{2}{2}} - 96)}{32^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 \sqrt{2} (16 \cdot 2^{\frac{2}{2}} - 96)}{32^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{8 \sqrt{2} (16 \cdot 2^{1} - 96)}{32^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{8 \sqrt{2} (16 \cdot 2 - 96)}{32^{\frac{3}{2}}}$

Multiply $16$ by $2$ .

$\frac{8 \sqrt{2} (32 - 96)}{32^{\frac{3}{2}}}$

Subtract $96$ from $32$ .

$\frac{8 \sqrt{2} \cdot - 64}{32^{\frac{3}{2}}}$

Multiply $- 64$ by $8$ .

$\frac{- 512 \sqrt{2}}{32^{\frac{3}{2}}}$

Move the negative in front of the fraction.

$- \frac{512 \sqrt{2}}{32^{\frac{3}{2}}}$

Step 11

$x = 4 \sqrt{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 4 \sqrt{2}$ is a local maximum

Step 12

Find the y-value when $x = 4 \sqrt{2}$ .

Replace the variable $x$ with $4 \sqrt{2}$ in the expression.

$f (4 \sqrt{2}) = (4 \sqrt{2}) \sqrt{64 - {(4 \sqrt{2})}^{2}}$

Simplify the result.

Simplify the expression.

Apply the product rule to $4 \sqrt{2}$ .

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (4^{2} {\sqrt{2}}^{2})}$

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

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 {\sqrt{2}}^{2})}$

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

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

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 {(2^{\frac{1}{2}})}^{2})}$

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

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{1}{2} \cdot 2})}$

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

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{2}{2}})}$

Rewrite the expression.

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 \cdot 2)}$

Evaluate the exponent.

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - (16 \cdot 2)}$

Multiply $- (16 \cdot 2)$ .

Multiply $16$ by $2$ .

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - 1 \cdot 32}$

Multiply $- 1$ by $32$ .

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{64 - 32}$

Subtract $32$ from $64$ .

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{32}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$f (4 \sqrt{2}) = 4 \sqrt{2} \sqrt{16 (2)}$

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

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

Pull terms out from under the radical.

$f (4 \sqrt{2}) = 4 \sqrt{2} (4 \sqrt{2})$

Multiply $4 \sqrt{2} (4 \sqrt{2})$ .

Multiply $4$ by $4$ .

$f (4 \sqrt{2}) = 16 \sqrt{2} \sqrt{2}$

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

$f (4 \sqrt{2}) = 16 (\sqrt{2} \sqrt{2})$

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

$f (4 \sqrt{2}) = 16 (\sqrt{2} \sqrt{2})$

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

$f (4 \sqrt{2}) = 16 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$f (4 \sqrt{2}) = 16 {\sqrt{2}}^{2}$

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

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

$f (4 \sqrt{2}) = 16 {(2^{\frac{1}{2}})}^{2}$

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

$f (4 \sqrt{2}) = 16 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$f (4 \sqrt{2}) = 16 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (4 \sqrt{2}) = 16 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$f (4 \sqrt{2}) = 16 \cdot 2$

Evaluate the exponent.

$f (4 \sqrt{2}) = 16 \cdot 2$

Multiply $16$ by $2$ .

$f (4 \sqrt{2}) = 32$

The final answer is $32$ .

$y = 32$

Step 13

Evaluate the second derivative at $x = - 4 \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (- 4 \sqrt{2}) ({(- 4 \sqrt{2})}^{2} - 96)}{{(- {(- 4 \sqrt{2})}^{2} + 64)}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

Multiply $- 4$ by $2$ .

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- {(- 4 \sqrt{2})}^{2} + 64)}^{\frac{3}{2}}}$

Simplify the denominator.

Simplify each term.

Apply the product rule to $- 4 \sqrt{2}$ .

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- ({(- 4)}^{2} {\sqrt{2}}^{2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 {\sqrt{2}}^{2}) + 64)}^{\frac{3}{2}}}$

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

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

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 {(2^{\frac{1}{2}})}^{2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{1}{2} \cdot 2}) + 64)}^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{2}{2}}) + 64)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{\frac{2}{2}}) + 64)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2^{1}) + 64)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- (16 \cdot 2) + 64)}^{\frac{3}{2}}}$

Multiply $- (16 \cdot 2)$ .

Multiply $16$ by $2$ .

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- 1 \cdot 32 + 64)}^{\frac{3}{2}}}$

Multiply $- 1$ by $32$ .

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{{(- 32 + 64)}^{\frac{3}{2}}}$

Add $- 32$ and $64$ .

$\frac{- 8 \sqrt{2} ({(- 4 \sqrt{2})}^{2} - 96)}{32^{\frac{3}{2}}}$

Simplify the numerator.

Apply the product rule to $- 4 \sqrt{2}$ .

$\frac{- 8 \sqrt{2} ({(- 4)}^{2} {\sqrt{2}}^{2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} (16 {\sqrt{2}}^{2} - 96)}{32^{\frac{3}{2}}}$

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

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

$\frac{- 8 \sqrt{2} (16 {(2^{\frac{1}{2}})}^{2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} (16 \cdot 2^{\frac{1}{2} \cdot 2} - 96)}{32^{\frac{3}{2}}}$

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

$\frac{- 8 \sqrt{2} (16 \cdot 2^{\frac{2}{2}} - 96)}{32^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- 8 \sqrt{2} (16 \cdot 2^{\frac{2}{2}} - 96)}{32^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{- 8 \sqrt{2} (16 \cdot 2^{1} - 96)}{32^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{- 8 \sqrt{2} (16 \cdot 2 - 96)}{32^{\frac{3}{2}}}$

Multiply $16$ by $2$ .

$\frac{- 8 \sqrt{2} (32 - 96)}{32^{\frac{3}{2}}}$

Subtract $96$ from $32$ .

$\frac{- 8 \sqrt{2} \cdot - 64}{32^{\frac{3}{2}}}$

Multiply $- 64$ by $- 8$ .

$\frac{512 \sqrt{2}}{32^{\frac{3}{2}}}$

Step 15

$x = - 4 \sqrt{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 4 \sqrt{2}$ is a local minimum

Step 16

Find the y-value when $x = - 4 \sqrt{2}$ .

Replace the variable $x$ with $- 4 \sqrt{2}$ in the expression.

$f (- 4 \sqrt{2}) = (- 4 \sqrt{2}) \sqrt{64 - {(- 4 \sqrt{2})}^{2}}$

Simplify the result.

Simplify the expression.

Apply the product rule to $- 4 \sqrt{2}$ .

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - ({(- 4)}^{2} {\sqrt{2}}^{2})}$

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

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 {\sqrt{2}}^{2})}$

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

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

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 {(2^{\frac{1}{2}})}^{2})}$

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

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{1}{2} \cdot 2})}$

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

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 \cdot 2^{\frac{2}{2}})}$

Rewrite the expression.

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 \cdot 2)}$

Evaluate the exponent.

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - (16 \cdot 2)}$

Multiply $- (16 \cdot 2)$ .

Multiply $16$ by $2$ .

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - 1 \cdot 32}$

Multiply $- 1$ by $32$ .

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{64 - 32}$

Subtract $32$ from $64$ .

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{32}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$f (- 4 \sqrt{2}) = - 4 \sqrt{2} \sqrt{16 (2)}$

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

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

Pull terms out from under the radical.

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

Multiply $- 4 \sqrt{2} (4 \sqrt{2})$ .

Multiply $4$ by $- 4$ .

$f (- 4 \sqrt{2}) = - 16 \sqrt{2} \sqrt{2}$

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

$f (- 4 \sqrt{2}) = - 16 (\sqrt{2} \sqrt{2})$

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

$f (- 4 \sqrt{2}) = - 16 (\sqrt{2} \sqrt{2})$

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

$f (- 4 \sqrt{2}) = - 16 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$f (- 4 \sqrt{2}) = - 16 {\sqrt{2}}^{2}$

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

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

$f (- 4 \sqrt{2}) = - 16 {(2^{\frac{1}{2}})}^{2}$

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

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

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

$f (- 4 \sqrt{2}) = - 16 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 4 \sqrt{2}) = - 16 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$f (- 4 \sqrt{2}) = - 16 \cdot 2$

Evaluate the exponent.

$f (- 4 \sqrt{2}) = - 16 \cdot 2$

Multiply $- 16$ by $2$ .

$f (- 4 \sqrt{2}) = - 32$

The final answer is $- 32$ .

$y = - 32$

Step 17

Evaluate the second derivative at $x = - 8$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{{(- {(- 8)}^{2} + 64)}^{\frac{3}{2}}}$

Step 18

Evaluate the second derivative.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{{(- 1 \cdot 64 + 64)}^{\frac{3}{2}}}$

Multiply $- 1$ by $64$ .

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{{(- 64 + 64)}^{\frac{3}{2}}}$

Reduce the expression by cancelling the common factors.

Add $- 64$ and $64$ .

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0^{\frac{3}{2}}}$

Simplify the expression.

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

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{{(0^{2})}^{\frac{3}{2}}}$

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

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0^{2 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0^{2 (\frac{3}{2})}}$

Rewrite the expression.

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0^{3}}$

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

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0}$

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

Undefined

$\frac{2 (- 8) ({(- 8)}^{2} - 96)}{0}$

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 x square root of 64-x^2

Related Questions

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