Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

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

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

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $2$ by $2$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $- 8$ by $1$ .

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

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

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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.

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Step 2.2.1

Factor $4$ out of $4 x^{3} + 4 x - 8$ .

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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.

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Step 2.2.2.1

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

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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.

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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 $x - 1$ .

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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 $x$ .

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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 $x$ .

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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 $x$ .

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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.

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Step 2.5.2.3.1

Simplify the numerator.

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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 $- 4 \cdot 1 \cdot 2$ .

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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 $\pm$ .

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Step 2.5.2.4.1

Simplify the numerator.

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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 $- 4 \cdot 1 \cdot 2$ .

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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 $\pm$ .

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Step 2.5.2.5.1

Simplify the numerator.

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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 $- 4 \cdot 1 \cdot 2$ .

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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.

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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.

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Step 5.2.1

Simplify each term.

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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.

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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

At $x = 0$ the derivative is $- 8$ . Since this is negative, the function is decreasing on $(- \infty, 1)$ .

Decreasing on $(- \infty, 1)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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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.

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Step 6.2.1

Simplify each term.

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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.

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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

At $x = 2$ the derivative is $32$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 7

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

Increasing on: $(1, \infty)$

Decreasing on: $(- \infty, 1)$

Step 8

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