Calculus Examples

$4 x^{3} - 256 x$

Step 1

Write $4 x^{3} - 256 x$ as a function.

$f (x) = 4 x^{3} - 256 x$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $4 x^{3} - 256 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 256 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 256 x]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Tap for more steps...

Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 256 x]$

Step 2.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 256 x]$

Step 2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 256 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 256 x]$ .

Tap for more steps...

Step 2.3.1

Since $- 256$ is constant with respect to $x$ , the derivative of $- 256 x$ with respect to $x$ is $- 256 \frac{d}{d x} [x]$ .

$12 x^{2} - 256 \frac{d}{d x} [x]$

Step 2.3.2

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

$12 x^{2} - 256 \cdot 1$

Step 2.3.3

Multiply $- 256$ by $1$ .

$12 x^{2} - 256$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $12 x^{2} - 256$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 256]$ .

$f'' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 256)$

Step 3.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Tap for more steps...

Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 256)$

Step 3.2.2

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

$f'' (x) = 12 (2 x) + \frac{d}{d x} (- 256)$

Step 3.2.3

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (- 256)$

Step 3.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.3.1

Since $- 256$ is constant with respect to $x$ , the derivative of $- 256$ with respect to $x$ is $0$ .

$f'' (x) = 24 x + 0$

Step 3.3.2

Add $24 x$ and $0$ .

$f'' (x) = 24 x$

Step 4

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

$12 x^{2} - 256 = 0$

Step 5

Find the first derivative.

Tap for more steps...

Step 5.1

Find the first derivative.

Tap for more steps...

Step 5.1.1

By the Sum Rule, the derivative of $4 x^{3} - 256 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 256 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 256 x]$

Step 5.1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Tap for more steps...

Step 5.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 256 x]$

Step 5.1.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 256 x]$

Step 5.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 256 x]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 256 x]$ .

Tap for more steps...

Step 5.1.3.1

Since $- 256$ is constant with respect to $x$ , the derivative of $- 256 x$ with respect to $x$ is $- 256 \frac{d}{d x} [x]$ .

$12 x^{2} - 256 \frac{d}{d x} [x]$

Step 5.1.3.2

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

$12 x^{2} - 256 \cdot 1$

Step 5.1.3.3

Multiply $- 256$ by $1$ .

$f' (x) = 12 x^{2} - 256$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 256$ .

$12 x^{2} - 256$

Step 6

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 256 = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

$12 x^{2} - 256 = 0$

Step 6.2

Add $256$ to both sides of the equation.

$12 x^{2} = 256$

Step 6.3

Divide each term in $12 x^{2} = 256$ by $12$ and simplify.

Tap for more steps...

Step 6.3.1

Divide each term in $12 x^{2} = 256$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{256}{12}$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{256}{12}$

Step 6.3.2.1.2

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

$x^{2} = \frac{256}{12}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Cancel the common factor of $256$ and $12$ .

Tap for more steps...

Step 6.3.3.1.1

Factor $4$ out of $256$ .

$x^{2} = \frac{4 (64)}{12}$

Step 6.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.3.3.1.2.1

Factor $4$ out of $12$ .

$x^{2} = \frac{4 \cdot 64}{4 \cdot 3}$

Step 6.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{4 \cdot 64}{4 \cdot 3}$

Step 6.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{64}{3}$

Step 6.4

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

$x = \pm \sqrt{\frac{64}{3}}$

Step 6.5

Simplify $\pm \sqrt{\frac{64}{3}}$ .

Tap for more steps...

Step 6.5.1

Rewrite $\sqrt{\frac{64}{3}}$ as $\frac{\sqrt{64}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{64}}{\sqrt{3}}$

Step 6.5.2

Simplify the numerator.

Tap for more steps...

Step 6.5.2.1

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

$x = \pm \frac{\sqrt{8^{2}}}{\sqrt{3}}$

Step 6.5.2.2

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

$x = \pm \frac{8}{\sqrt{3}}$

Step 6.5.3

Multiply $\frac{8}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{8}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 6.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 6.5.4.1

Multiply $\frac{8}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{8 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{8 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 6.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{8 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 6.5.4.4

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

$x = \pm \frac{8 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{8 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 6.5.4.6

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

Tap for more steps...

Step 6.5.4.6.1

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

$x = \pm \frac{8 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 6.5.4.6.2

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

$x = \pm \frac{8 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 6.5.4.6.3

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

$x = \pm \frac{8 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{8 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{8 \sqrt{3}}{3^{1}}$

Step 6.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{8 \sqrt{3}}{3}$

Step 6.6

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

Tap for more steps...

Step 6.6.1

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

$x = \frac{8 \sqrt{3}}{3}$

Step 6.6.2

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

$x = - \frac{8 \sqrt{3}}{3}$

Step 6.6.3

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

$x = \frac{8 \sqrt{3}}{3}, - \frac{8 \sqrt{3}}{3}$

Step 7

Find the values where the derivative is undefined.

Tap for more steps...

Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = \frac{8 \sqrt{3}}{3}, - \frac{8 \sqrt{3}}{3}$

Step 9

Evaluate the second derivative at $x = \frac{8 \sqrt{3}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (\frac{8 \sqrt{3}}{3})$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 10.1.1

Factor $3$ out of $24$ .

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

Step 10.1.2

Cancel the common factor.

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

Step 10.1.3

Rewrite the expression.

$8 (8 \sqrt{3})$

Step 10.2

Multiply $8$ by $8$ .

$64 \sqrt{3}$

Step 11

$x = \frac{8 \sqrt{3}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{8 \sqrt{3}}{3}$ is a local minimum

Step 12

Find the y-value when $x = \frac{8 \sqrt{3}}{3}$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $\frac{8 \sqrt{3}}{3}$ in the expression.

$f (\frac{8 \sqrt{3}}{3}) = 4 {(\frac{8 \sqrt{3}}{3})}^{3} - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

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

Tap for more steps...

Step 12.2.1.1.1

Apply the product rule to $\frac{8 \sqrt{3}}{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{{(8 \sqrt{3})}^{3}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.1.2

Apply the product rule to $8 \sqrt{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{8^{3} {\sqrt{3}}^{3}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2

Simplify the numerator.

Tap for more steps...

Step 12.2.1.2.1

Raise $8$ to the power of $3$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 {\sqrt{3}}^{3}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.2

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

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \sqrt{3^{3}}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.3

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

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \sqrt{27}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.4

Rewrite $27$ as $3^{2} \cdot 3$ .

Tap for more steps...

Step 12.2.1.2.4.1

Factor $9$ out of $27$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \sqrt{9 (3)}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.4.2

Rewrite $9$ as $3^{2}$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \sqrt{3^{2} \cdot 3}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.5

Pull terms out from under the radical.

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \cdot (3 \sqrt{3})}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.2.6

Multiply $512$ by $3$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{1536 \sqrt{3}}{3^{3}}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.3

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

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{1536 \sqrt{3}}{27}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.4

Cancel the common factor of $1536$ and $27$ .

Tap for more steps...

Step 12.2.1.4.1

Factor $3$ out of $1536 \sqrt{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{3 (512 \sqrt{3})}{27}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.4.2

Cancel the common factors.

Tap for more steps...

Step 12.2.1.4.2.1

Factor $3$ out of $27$ .

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{3 (512 \sqrt{3})}{3 (9)}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.4.2.2

Cancel the common factor.

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{3 (512 \sqrt{3})}{3 \cdot 9}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.4.2.3

Rewrite the expression.

$f (\frac{8 \sqrt{3}}{3}) = 4 (\frac{512 \sqrt{3}}{9}) - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.5

Multiply $4 \frac{512 \sqrt{3}}{9}$ .

Tap for more steps...

Step 12.2.1.5.1

Combine $4$ and $\frac{512 \sqrt{3}}{9}$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{4 (512 \sqrt{3})}{9} - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.5.2

Multiply $512$ by $4$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} - 256 \frac{8 \sqrt{3}}{3}$

Step 12.2.1.6

Multiply $- 256 \frac{8 \sqrt{3}}{3}$ .

Tap for more steps...

Step 12.2.1.6.1

Combine $- 256$ and $\frac{8 \sqrt{3}}{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} + \frac{- 256 (8 \sqrt{3})}{3}$

Step 12.2.1.6.2

Multiply $8$ by $- 256$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} + \frac{- 2048 \sqrt{3}}{3}$

Step 12.2.1.7

Move the negative in front of the fraction.

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} - \frac{2048 \sqrt{3}}{3}$

Step 12.2.2

To write $- \frac{2048 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} - \frac{2048 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 12.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 12.2.3.1

Multiply $\frac{2048 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} - \frac{2048 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 12.2.3.2

Multiply $3$ by $3$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3}}{9} - \frac{2048 \sqrt{3} \cdot 3}{9}$

Step 12.2.4

Combine the numerators over the common denominator.

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3} - 2048 \sqrt{3} \cdot 3}{9}$

Step 12.2.5

Simplify the numerator.

Tap for more steps...

Step 12.2.5.1

Multiply $3$ by $- 2048$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{2048 \sqrt{3} - 6144 \sqrt{3}}{9}$

Step 12.2.5.2

Subtract $6144 \sqrt{3}$ from $2048 \sqrt{3}$ .

$f (\frac{8 \sqrt{3}}{3}) = \frac{- 4096 \sqrt{3}}{9}$

Step 12.2.6

Move the negative in front of the fraction.

$f (\frac{8 \sqrt{3}}{3}) = - \frac{4096 \sqrt{3}}{9}$

Step 12.2.7

The final answer is $- \frac{4096 \sqrt{3}}{9}$ .

$y = - \frac{4096 \sqrt{3}}{9}$

Step 13

Evaluate the second derivative at $x = - \frac{8 \sqrt{3}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (- \frac{8 \sqrt{3}}{3})$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 14.1.1

Move the leading negative in $- \frac{8 \sqrt{3}}{3}$ into the numerator.

$24 \frac{- 8 \sqrt{3}}{3}$

Step 14.1.2

Factor $3$ out of $24$ .

$3 (8) \frac{- 8 \sqrt{3}}{3}$

Step 14.1.3

Cancel the common factor.

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

Step 14.1.4

Rewrite the expression.

$8 (- 8 \sqrt{3})$

Step 14.2

Multiply $- 8$ by $8$ .

$- 64 \sqrt{3}$

Step 15

$x = - \frac{8 \sqrt{3}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{8 \sqrt{3}}{3}$ is a local maximum

Step 16

Find the y-value when $x = - \frac{8 \sqrt{3}}{3}$ .

Tap for more steps...

Step 16.1

Replace the variable $x$ with $- \frac{8 \sqrt{3}}{3}$ in the expression.

$f (- \frac{8 \sqrt{3}}{3}) = 4 {(- \frac{8 \sqrt{3}}{3})}^{3} - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2

Simplify the result.

Tap for more steps...

Step 16.2.1

Simplify each term.

Tap for more steps...

Step 16.2.1.1

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

Tap for more steps...

Step 16.2.1.1.1

Apply the product rule to $- \frac{8 \sqrt{3}}{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 ({(- 1)}^{3} {(\frac{8 \sqrt{3}}{3})}^{3}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.1.2

Apply the product rule to $\frac{8 \sqrt{3}}{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 ({(- 1)}^{3} (\frac{{(8 \sqrt{3})}^{3}}{3^{3}})) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.1.3

Apply the product rule to $8 \sqrt{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 ({(- 1)}^{3} (\frac{8^{3} {\sqrt{3}}^{3}}{3^{3}})) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.2

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

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{8^{3} {\sqrt{3}}^{3}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3

Simplify the numerator.

Tap for more steps...

Step 16.2.1.3.1

Raise $8$ to the power of $3$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 {\sqrt{3}}^{3}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.2

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

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \sqrt{3^{3}}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.3

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

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \sqrt{27}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.4

Rewrite $27$ as $3^{2} \cdot 3$ .

Tap for more steps...

Step 16.2.1.3.4.1

Factor $9$ out of $27$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \sqrt{9 (3)}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.4.2

Rewrite $9$ as $3^{2}$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \sqrt{3^{2} \cdot 3}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.5

Pull terms out from under the radical.

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \cdot (3 \sqrt{3})}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.3.6

Multiply $512$ by $3$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{1536 \sqrt{3}}{3^{3}}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.4

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

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{1536 \sqrt{3}}{27}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.5

Cancel the common factor of $1536$ and $27$ .

Tap for more steps...

Step 16.2.1.5.1

Factor $3$ out of $1536 \sqrt{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{3 (512 \sqrt{3})}{27}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.5.2

Cancel the common factors.

Tap for more steps...

Step 16.2.1.5.2.1

Factor $3$ out of $27$ .

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{3 (512 \sqrt{3})}{3 (9)}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.5.2.2

Cancel the common factor.

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{3 (512 \sqrt{3})}{3 \cdot 9}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.5.2.3

Rewrite the expression.

$f (- \frac{8 \sqrt{3}}{3}) = 4 (- \frac{512 \sqrt{3}}{9}) - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.6

Multiply $4 (- \frac{512 \sqrt{3}}{9})$ .

Tap for more steps...

Step 16.2.1.6.1

Multiply $- 1$ by $4$ .

$f (- \frac{8 \sqrt{3}}{3}) = - 4 \frac{512 \sqrt{3}}{9} - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.6.2

Combine $- 4$ and $\frac{512 \sqrt{3}}{9}$ .

$f (- \frac{8 \sqrt{3}}{3}) = \frac{- 4 (512 \sqrt{3})}{9} - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.6.3

Multiply $512$ by $- 4$ .

$f (- \frac{8 \sqrt{3}}{3}) = \frac{- 2048 \sqrt{3}}{9} - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.7

Move the negative in front of the fraction.

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} - 256 (- \frac{8 \sqrt{3}}{3})$

Step 16.2.1.8

Multiply $- 256 (- \frac{8 \sqrt{3}}{3})$ .

Tap for more steps...

Step 16.2.1.8.1

Multiply $- 1$ by $- 256$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + 256 (\frac{8 \sqrt{3}}{3})$

Step 16.2.1.8.2

Combine $256$ and $\frac{8 \sqrt{3}}{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + \frac{256 (8 \sqrt{3})}{3}$

Step 16.2.1.8.3

Multiply $8$ by $256$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + \frac{2048 \sqrt{3}}{3}$

Step 16.2.2

To write $\frac{2048 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + \frac{2048 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 16.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 16.2.3.1

Multiply $\frac{2048 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + \frac{2048 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 16.2.3.2

Multiply $3$ by $3$ .

$f (- \frac{8 \sqrt{3}}{3}) = - \frac{2048 \sqrt{3}}{9} + \frac{2048 \sqrt{3} \cdot 3}{9}$

Step 16.2.4

Combine the numerators over the common denominator.

$f (- \frac{8 \sqrt{3}}{3}) = \frac{- 2048 \sqrt{3} + 2048 \sqrt{3} \cdot 3}{9}$

Step 16.2.5

Simplify the numerator.

Tap for more steps...

Step 16.2.5.1

Multiply $3$ by $2048$ .

$f (- \frac{8 \sqrt{3}}{3}) = \frac{- 2048 \sqrt{3} + 6144 \sqrt{3}}{9}$

Step 16.2.5.2

Add $- 2048 \sqrt{3}$ and $6144 \sqrt{3}$ .

$f (- \frac{8 \sqrt{3}}{3}) = \frac{4096 \sqrt{3}}{9}$

Step 16.2.6

The final answer is $\frac{4096 \sqrt{3}}{9}$ .

$y = \frac{4096 \sqrt{3}}{9}$

Step 17

These are the local extrema for $f (x) = 4 x^{3} - 256 x$ .

$(\frac{8 \sqrt{3}}{3}, - \frac{4096 \sqrt{3}}{9})$ is a local minima

$(- \frac{8 \sqrt{3}}{3}, \frac{4096 \sqrt{3}}{9})$ is a local maxima

Step 18