Calculus Examples

$f (x) = x + \frac{32}{x^{2}}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x + \frac{32}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{32}{x^{2}}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 1.1.2

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

$1 + \frac{d}{d x} [\frac{32}{x^{2}}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{32}{x^{2}}]$ .

Tap for more steps...

Step 1.2.1

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{2}}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$1 + 32 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.2.2

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

$1 + 32 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

$1 + 32 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.2.3.2

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

$1 + 32 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.2.3.3

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

$1 + 32 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.2.4

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

$1 + 32 (- {(x^{2})}^{- 2} (2 x))$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$1 + 32 (- x^{2 \cdot - 2} (2 x))$

Step 1.2.5.2

Multiply $2$ by $- 2$ .

$1 + 32 (- x^{- 4} (2 x))$

Step 1.2.6

Multiply $2$ by $- 1$ .

$1 + 32 (- 2 x^{- 4} x)$

Step 1.2.7

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

$1 + 32 (- 2 (x^{1} x^{- 4}))$

Step 1.2.8

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

$1 + 32 (- 2 x^{1 - 4})$

Step 1.2.9

Subtract $4$ from $1$ .

$1 + 32 (- 2 x^{- 3})$

Step 1.2.10

Multiply $- 2$ by $32$ .

$1 - 64 x^{- 3}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - 64 \frac{1}{x^{3}}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Combine terms.

Tap for more steps...

Step 1.4.1.1

Combine $- 64$ and $\frac{1}{x^{3}}$ .

$1 + \frac{- 64}{x^{3}}$

Step 1.4.1.2

Move the negative in front of the fraction.

$1 - \frac{64}{x^{3}}$

Step 1.4.2

Reorder terms.

$- \frac{64}{x^{3}} + 1$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- \frac{64}{x^{3}} + 1$ with respect to $x$ is $\frac{d}{d x} [- \frac{64}{x^{3}}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{d}{d x} (- \frac{64}{x^{3}}) + \frac{d}{d x} (1)$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{64}{x^{3}}]$ .

Tap for more steps...

Step 2.2.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- \frac{64}{x^{3}}$ with respect to $x$ is $- 64 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = - 64 \frac{d}{d x} \frac{1}{x^{3}} + \frac{d}{d x} (1)$

Step 2.2.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f'' (x) = - 64 \frac{d}{d x} {(x^{3})}^{- 1} + \frac{d}{d x} (1)$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$f'' (x) = - 64 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (1)$

Step 2.2.3.2

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

$f'' (x) = - 64 (- u^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (1)$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$f'' (x) = - 64 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (1)$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - 64 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (1)$

Step 2.2.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

Tap for more steps...

Step 2.2.5.1

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

$f'' (x) = - 64 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (1)$

Step 2.2.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = - 64 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (1)$

Step 2.2.6

Multiply $3$ by $- 1$ .

$f'' (x) = - 64 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (1)$

Step 2.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.2.7.1

Move $x^{2}$ .

$f'' (x) = - 64 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (1)$

Step 2.2.7.2

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

$f'' (x) = - 64 (- 3 x^{2 - 6}) + \frac{d}{d x} (1)$

Step 2.2.7.3

Subtract $6$ from $2$ .

$f'' (x) = - 64 (- 3 x^{- 4}) + \frac{d}{d x} (1)$

Step 2.2.8

Multiply $- 3$ by $- 64$ .

$f'' (x) = 192 x^{- 4} + \frac{d}{d x} (1)$

Step 2.3

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

$f'' (x) = 192 x^{- 4} + 0$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 192 (\frac{1}{x^{4}}) + 0$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Combine $192$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{192}{x^{4}} + 0$

Step 2.4.2.2

Add $\frac{192}{x^{4}}$ and $0$ .

$f'' (x) = \frac{192}{x^{4}}$

Step 3

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

$- \frac{64}{x^{3}} + 1 = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

Differentiate.

Tap for more steps...

Step 4.1.1.1

By the Sum Rule, the derivative of $x + \frac{32}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{32}{x^{2}}]$ .

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (\frac{32}{x^{2}})$

Step 4.1.1.2

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

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

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{32}{x^{2}}]$ .

Tap for more steps...

Step 4.1.2.1

Since $32$ is constant with respect to $x$ , the derivative of $\frac{32}{x^{2}}$ with respect to $x$ is $32 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

Tap for more steps...

Step 4.1.2.3.1

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

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

Step 4.1.2.3.2

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

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

Step 4.1.2.3.3

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

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

Step 4.1.2.4

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

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

Step 4.1.2.5

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

Tap for more steps...

Step 4.1.2.5.1

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

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

Step 4.1.2.5.2

Multiply $2$ by $- 2$ .

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

Step 4.1.2.6

Multiply $2$ by $- 1$ .

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

Step 4.1.2.7

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

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

Step 4.1.2.8

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

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

Step 4.1.2.9

Subtract $4$ from $1$ .

$f' (x) = 1 + 32 (- 2 x^{- 3})$

Step 4.1.2.10

Multiply $- 2$ by $32$ .

$f' (x) = 1 - 64 x^{- 3}$

Step 4.1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = 1 - 64 \frac{1}{x^{3}}$

Step 4.1.4

Simplify.

Tap for more steps...

Step 4.1.4.1

Combine terms.

Tap for more steps...

Step 4.1.4.1.1

Combine $- 64$ and $\frac{1}{x^{3}}$ .

$f' (x) = 1 + \frac{- 64}{x^{3}}$

Step 4.1.4.1.2

Move the negative in front of the fraction.

$f' (x) = 1 - \frac{64}{x^{3}}$

Step 4.1.4.2

Reorder terms.

$f' (x) = - \frac{64}{x^{3}} + 1$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{64}{x^{3}} + 1$ .

$- \frac{64}{x^{3}} + 1$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{64}{x^{3}} + 1 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- \frac{64}{x^{3}} + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$- \frac{64}{x^{3}} = - 1$

Step 5.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.3.1

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

$x^{3}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 5.4

Multiply each term in $- \frac{64}{x^{3}} = - 1$ by $x^{3}$ to eliminate the fractions.

Tap for more steps...

Step 5.4.1

Multiply each term in $- \frac{64}{x^{3}} = - 1$ by $x^{3}$ .

$- \frac{64}{x^{3}} x^{3} = - x^{3}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $x^{3}$ .

Tap for more steps...

Step 5.4.2.1.1

Move the leading negative in $- \frac{64}{x^{3}}$ into the numerator.

$\frac{- 64}{x^{3}} x^{3} = - x^{3}$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- 64}{x^{3}} x^{3} = - x^{3}$

Step 5.4.2.1.3

Rewrite the expression.

$- 64 = - x^{3}$

Step 5.5

Solve the equation.

Tap for more steps...

Step 5.5.1

Rewrite the equation as $- x^{3} = - 64$ .

$- x^{3} = - 64$

Step 5.5.2

Add $64$ to both sides of the equation.

$- x^{3} + 64 = 0$

Step 5.5.3

Factor the left side of the equation.

Tap for more steps...

Step 5.5.3.1

Factor $- 1$ out of $- x^{3} + 64$ .

Tap for more steps...

Step 5.5.3.1.1

Factor $- 1$ out of $- x^{3}$ .

$- (x^{3}) + 64 = 0$

Step 5.5.3.1.2

Rewrite $64$ as $- 1 (- 64)$ .

$- (x^{3}) - 1 \cdot - 64 = 0$

Step 5.5.3.1.3

Factor $- 1$ out of $- (x^{3}) - 1 (- 64)$ .

$- (x^{3} - 64) = 0$

Step 5.5.3.2

Rewrite $64$ as $4^{3}$ .

$- (x^{3} - 4^{3}) = 0$

Step 5.5.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

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

Step 5.5.3.4

Factor.

Tap for more steps...

Step 5.5.3.4.1

Simplify.

Tap for more steps...

Step 5.5.3.4.1.1

Move $4$ to the left of $x$ .

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

Step 5.5.3.4.1.2

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

$- ((x - 4) (x^{2} + 4 x + 16)) = 0$

Step 5.5.3.4.2

Remove unnecessary parentheses.

$- (x - 4) (x^{2} + 4 x + 16) = 0$

Step 5.5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x^{2} + 4 x + 16 = 0$

Step 5.5.5

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 5.5.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.5.6

Set $x^{2} + 4 x + 16$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.6.1

Set $x^{2} + 4 x + 16$ equal to $0$ .

$x^{2} + 4 x + 16 = 0$

Step 5.5.6.2

Solve $x^{2} + 4 x + 16 = 0$ for $x$ .

Tap for more steps...

Step 5.5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.5.6.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

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

Step 5.5.6.2.3

Simplify.

Tap for more steps...

Step 5.5.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.5.6.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 5.5.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 5.5.6.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 5.5.6.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 5.5.6.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.3.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 5.5.6.2.3.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 5.5.6.2.3.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 5.5.6.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 5.5.6.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 5.5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 5.5.6.2.4.1

Simplify the numerator.

Tap for more steps...

Step 5.5.6.2.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 5.5.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.4.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 5.5.6.2.4.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 5.5.6.2.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 5.5.6.2.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 5.5.6.2.4.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 5.5.6.2.4.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.6.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 5.5.6.2.4.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 5.5.6.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 5.5.6.2.4.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 i \sqrt{3}$

Step 5.5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 5.5.6.2.5.1

Simplify the numerator.

Tap for more steps...

Step 5.5.6.2.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 5.5.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 5.5.6.2.5.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 5.5.6.2.5.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 5.5.6.2.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 5.5.6.2.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.5.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 5.5.6.2.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 5.5.6.2.5.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 5.5.6.2.5.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.6.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 5.5.6.2.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 5.5.6.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 5.5.6.2.5.4

Change the $\pm$ to $-$ .

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

Step 5.5.6.2.6

The final answer is the combination of both solutions.

$x = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 5.5.7

The final solution is all the values that make $- (x - 4) (x^{2} + 4 x + 16) = 0$ true.

$x = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

Set the denominator in $\frac{64}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 6.2

Solve for $x$ .

Tap for more steps...

Step 6.2.1

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

$x = \sqrt[3]{0}$

Step 6.2.2

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 6.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.2.2.2

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

$x = 0$

Step 7

Critical points to evaluate.

$x = 4$

Step 8

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

$\frac{192}{{(4)}^{4}}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

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

$\frac{192}{256}$

Step 9.2

Cancel the common factor of $192$ and $256$ .

Tap for more steps...

Step 9.2.1

Factor $64$ out of $192$ .

$\frac{64 (3)}{256}$

Step 9.2.2

Cancel the common factors.

Tap for more steps...

Step 9.2.2.1

Factor $64$ out of $256$ .

$\frac{64 \cdot 3}{64 \cdot 4}$

Step 9.2.2.2

Cancel the common factor.

$\frac{64 \cdot 3}{64 \cdot 4}$

Step 9.2.2.3

Rewrite the expression.

$\frac{3}{4}$

Step 10

$x = 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4$ is a local minimum

Step 11

Find the y-value when $x = 4$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = (4) + \frac{32}{{(4)}^{2}}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

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

$f (4) = 4 + \frac{32}{16}$

Step 11.2.1.2

Divide $32$ by $16$ .

$f (4) = 4 + 2$

Step 11.2.2

Add $4$ and $2$ .

$f (4) = 6$

Step 11.2.3

The final answer is $6$ .

$y = 6$

Step 12

These are the local extrema for $f (x) = x + \frac{32}{x^{2}}$ .

$(4, 6)$ is a local minima

Step 13