Calculus Examples

f (x) = x^{4} + 2 x^{2} - 8 x

$f (x) = x^{4} + 2 x^{2} - 8 x$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of

x^{4} + 2 x^{2} - 8 x

$x^{4} + 2 x^{2} - 8 x$ with respect to

x

$x$ is

\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]$ .

\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]$

Step 1.1.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 4

$n = 4$ .

4 x^{3} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]

$4 x^{3} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]$

4 x^{3} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]

$4 x^{3} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x]$

Step 1.1.2

Evaluate

\frac{d}{d x} [2 x^{2}]

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

Tap for more steps...

Step 1.1.2.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x^{2}

$2 x^{2}$ with respect to

x

$x$ is

2 \frac{d}{d x} [x^{2}]

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

4 x^{3} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x]

$4 x^{3} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x]$

Step 1.1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

4 x^{3} + 2 (2 x) + \frac{d}{d x} [- 8 x]

$4 x^{3} + 2 (2 x) + \frac{d}{d x} [- 8 x]$

Step 1.1.2.3

Multiply

2

$2$ by

2

$2$ .

4 x^{3} + 4 x + \frac{d}{d x} [- 8 x]

$4 x^{3} + 4 x + \frac{d}{d x} [- 8 x]$

4 x^{3} + 4 x + \frac{d}{d x} [- 8 x]

$4 x^{3} + 4 x + \frac{d}{d x} [- 8 x]$

Step 1.1.3

Evaluate

\frac{d}{d x} [- 8 x]

$\frac{d}{d x} [- 8 x]$ .

Tap for more steps...

Step 1.1.3.1

Since

- 8

$- 8$ is constant with respect to

x

$x$ , the derivative of

- 8 x

$- 8 x$ with respect to

x

$x$ is

- 8 \frac{d}{d x} [x]

$- 8 \frac{d}{d x} [x]$ .

4 x^{3} + 4 x - 8 \frac{d}{d x} [x]

$4 x^{3} + 4 x - 8 \frac{d}{d x} [x]$

Step 1.1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

4 x^{3} + 4 x - 8 \cdot 1

$4 x^{3} + 4 x - 8 \cdot 1$

Step 1.1.3.3

Multiply

- 8

$- 8$ by

1

$1$ .

$f' (x) = 4 x^{3} + 4 x - 8$

Step 1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$4 x^{3} + 4 x - 8$ .

$4 x^{3} + 4 x - 8$

Step 2

Set the first derivative equal to

$0$ then solve the equation

$4 x^{3} + 4 x - 8 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to

$0$ .

$4 x^{3} + 4 x - 8 = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor

$4$ out of

$4 x^{3} + 4 x - 8$ .

Tap for more steps...

Step 2.2.1.1

Factor

$4$ out of

$4 x^{3}$ .

$4 (x^{3}) + 4 x - 8 = 0$

Step 2.2.1.2

Factor

$4$ out of

$4 x$ .

$4 (x^{3}) + 4 (x) - 8 = 0$

Step 2.2.1.3

Factor

$4$ out of

$- 8$ .

$4 (x^{3}) + 4 x + 4 \cdot - 2 = 0$

Step 2.2.1.4

Factor

$4$ out of

$4 (x^{3}) + 4 x$ .

$4 (x^{3} + x) + 4 \cdot - 2 = 0$

Step 2.2.1.5

Factor

$4$ out of

$4 (x^{3} + x) + 4 \cdot - 2$ .

$4 (x^{3} + x - 2) = 0$

Step 2.2.2

Factor.

Tap for more steps...

Step 2.2.2.1

Factor

$x^{3} + x - 2$ using the rational roots test.

Tap for more steps...

Step 2.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form

$\frac{p}{q}$ where

$p$ is a factor of the constant and

$q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 2.2.2.1.2

Find every combination of

$\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2$

Step 2.2.2.1.3

Substitute

$1$ and simplify the expression. In this case, the expression is equal to

$0$ so

$1$ is a root of the polynomial.

Tap for more steps...

Step 2.2.2.1.3.1

Substitute

$1$ into the polynomial.

$1^{3} + 1 - 2$

Step 2.2.2.1.3.2

Raise

$1$ to the power of

$3$ .

$1 + 1 - 2$

Step 2.2.2.1.3.3

Add

$1$ and

$1$ .

$2 - 2$

Step 2.2.2.1.3.4

Subtract

$2$ from

$2$ .

$0$

Step 2.2.2.1.4

Since

$1$ is a known root, divide the polynomial by

$x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + x - 2}{x - 1}$

Step 2.2.2.1.5

Divide

$x^{3} + x - 2$ by

$x - 1$ .

Tap for more steps...

Step 2.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of

$0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$x$

$2$

Step 2.2.2.1.5.2

Divide the highest order term in the dividend

$x^{3}$ by the highest order term in divisor

$x$ .

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			+	$x^{3}$	-	$x^{2}$

Step 2.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

$x^{3} - x^{2}$

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$

Step 2.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$

Step 2.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend

$x^{2}$ by the highest order term in divisor

$x$ .

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					+	$x^{2}$	-	$x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$x^{2} - x$

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$

Step 2.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$

Step 2.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$	-	$2$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend

$2 x$ by the highest order term in divisor

$x$ .

				$x^{2}$	+	$x$	+	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$	-	$2$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	+	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$	-	$2$
							+	$2 x$	-	$2$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$2 x - 2$

				$x^{2}$	+	$x$	+	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$	-	$2$
							-	$2 x$	+	$2$

Step 2.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$x$	+	$2$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	+	$x$	-	$2$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$x$
					-	$x^{2}$	+	$x$
							+	$2 x$	-	$2$
							-	$2 x$	+	$2$
										$0$

Step 2.2.2.1.5.16

Since the remander is

$0$ , the final answer is the quotient.

$x^{2} + x + 2$

Step 2.2.2.1.6

Write

$x^{3} + x - 2$ as a set of factors.

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

Step 2.2.2.2

Remove unnecessary parentheses.

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

Step 2.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x - 1 = 0$

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

Step 2.4

Set

$x - 1$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 2.4.1

Set

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 2.4.2

Add

$1$ to both sides of the equation.

$x = 1$

Step 2.5

Set

$x^{2} + x + 2$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 2.5.1

Set

$x^{2} + x + 2$ equal to

$0$ .

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

Step 2.5.2

Solve

$x^{2} + x + 2 = 0$ for

$x$ .

Tap for more steps...

Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values

$a = 1$ ,

$b = 1$ , and

$c = 2$ into the quadratic formula and solve for

$x$ .

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

Step 2.5.2.3

Simplify.

Tap for more steps...

Step 2.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.3.1.1

One to any power is one.

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

Step 2.5.2.3.1.2

Multiply

$- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.5.2.3.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 2.5.2.3.1.2.2

Multiply

$- 4$ by

$2$ .

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

Step 2.5.2.3.1.3

Subtract

$8$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite

$- 7$ as

$- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Rewrite

$\sqrt{- 1 (7)}$ as

$\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply

$2$ by

$1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 2.5.2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

Tap for more steps...

Step 2.5.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.4.1.1

One to any power is one.

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

Step 2.5.2.4.1.2

Multiply

$- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.5.2.4.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 2.5.2.4.1.2.2

Multiply

$- 4$ by

$2$ .

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

Step 2.5.2.4.1.3

Subtract

$8$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 2.5.2.4.1.4

Rewrite

$- 7$ as

$- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 2.5.2.4.1.5

Rewrite

$\sqrt{- 1 (7)}$ as

$\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.4.2

Multiply

$2$ by

$1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 2.5.2.4.3

Change the

$\pm$ to

$+$ .

$x = \frac{- 1 + i \sqrt{7}}{2}$

Step 2.5.2.4.4

Rewrite

$- 1$ as

$- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{7}}{2}$

Step 2.5.2.4.5

Factor

$- 1$ out of

$i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{7})}{2}$

Step 2.5.2.4.6

Factor

$- 1$ out of

$- 1 (1) - (- i \sqrt{7})$ .

$x = \frac{- 1 (1 - i \sqrt{7})}{2}$

Step 2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{7}}{2}$

Step 2.5.2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

Tap for more steps...

Step 2.5.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.5.1.1

One to any power is one.

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

Step 2.5.2.5.1.2

Multiply

$- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.5.2.5.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 2.5.2.5.1.2.2

Multiply

$- 4$ by

$2$ .

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

Step 2.5.2.5.1.3

Subtract

$8$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 2.5.2.5.1.4

Rewrite

$- 7$ as

$- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 2.5.2.5.1.5

Rewrite

$\sqrt{- 1 (7)}$ as

$\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 2.5.2.5.2

Multiply

$2$ by

$1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 2.5.2.5.3

Change the

$\pm$ to

$-$ .

$x = \frac{- 1 - i \sqrt{7}}{2}$

Step 2.5.2.5.4

Rewrite

$- 1$ as

$- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{7}}{2}$

Step 2.5.2.5.5

Factor

$- 1$ out of

$- i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{7})}{2}$

Step 2.5.2.5.6

Factor

$- 1$ out of

$- 1 (1) - (i \sqrt{7})$ .

$x = \frac{- 1 (1 + i \sqrt{7})}{2}$

Step 2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{7}}{2}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{7}}{2}, - \frac{1 + i \sqrt{7}}{2}$

Step 2.6

The final solution is all the values that make

$4 (x - 1) (x^{2} + x + 2) = 0$ true.

$x = 1, - \frac{1 - i \sqrt{7}}{2}, - \frac{1 + i \sqrt{7}}{2}$

Step 3

The values which make the derivative equal to

$0$ are

$1$ .

$1$

Step 4

After finding the point that makes the derivative

$f' (x) = 4 x^{3} + 4 x - 8$ equal to

$0$ or undefined, the interval to check where

$f (x) = x^{4} + 2 x^{2} - 8 x$ is increasing and where it is decreasing is

$(- \infty, 1) \cup (1, \infty)$ .

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

Step 5

Substitute a value from the interval

$(- \infty, 1)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable

$x$ with

$0$ in the expression.

$f' (0) = 4 {(0)}^{3} + 4 (0) - 8$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raising

$0$ to any positive power yields

$0$ .

$f' (0) = 4 \cdot 0 + 4 (0) - 8$

Step 5.2.1.2

Multiply

$4$ by

$0$ .

$f' (0) = 0 + 4 (0) - 8$

Step 5.2.1.3

Multiply

$4$ by

$0$ .

$f' (0) = 0 + 0 - 8$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Add

$0$ and

$0$ .

$f' (0) = 0 - 8$

Step 5.2.2.2

Subtract

$8$ from

$0$ .

$f' (0) = - 8$

Step 5.2.3

The final answer is

$- 8$ .

$- 8$

Step 5.3

$x = 0$ the derivative is

$- 8$ . Since this is negative, the function is decreasing on

$(- \infty, 1)$ .

Decreasing on

$(- \infty, 1)$ since

$f' (x) < 0$

Decreasing on

$(- \infty, 1)$ since

$f' (x) < 0$

Step 6

Substitute a value from the interval

$(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable

$x$ with

$2$ in the expression.

$f' (2) = 4 {(2)}^{3} + 4 (2) - 8$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise

$2$ to the power of

$3$ .

$f' (2) = 4 \cdot 8 + 4 (2) - 8$

Step 6.2.1.2

Multiply

$4$ by

$8$ .

$f' (2) = 32 + 4 (2) - 8$

Step 6.2.1.3

Multiply

$4$ by

$2$ .

$f' (2) = 32 + 8 - 8$

Step 6.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 6.2.2.1

Add

$32$ and

$8$ .

$f' (2) = 40 - 8$

Step 6.2.2.2

Subtract

$8$ from

$40$ .

$f' (2) = 32$

Step 6.2.3

The final answer is

$32$ .

$32$

Step 6.3

$x = 2$ the derivative is

$32$ . Since this is positive, the function is increasing on

$(1, \infty)$ .

Increasing on

$(1, \infty)$ since

$f' (x) > 0$

Increasing on

$(1, \infty)$ since

$f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on:

$(1, \infty)$

Decreasing on:

$(- \infty, 1)$

Step 8

Enter YOUR Problem