Calculus Examples

$f (x) = 3 x^{4} - 20 x^{3} + 42 x^{2} - 36 x$

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

By the Sum Rule, the derivative of $3 x^{4} - 20 x^{3} + 42 x^{2} - 36 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 20 x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 20 x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.2

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

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Step 1.1.2.1

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

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 20 x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

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 $n = 4$ .

$3 (4 x^{3}) + \frac{d}{d x} [- 20 x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 20 x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 20 x^{3}]$ .

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Step 1.1.3.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x^{3}$ with respect to $x$ is $- 20 \frac{d}{d x} [x^{3}]$ .

$12 x^{3} - 20 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

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 $n = 3$ .

$12 x^{3} - 20 (3 x^{2}) + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.3.3

Multiply $3$ by $- 20$ .

$12 x^{3} - 60 x^{2} + \frac{d}{d x} [42 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.4

Evaluate $\frac{d}{d x} [42 x^{2}]$ .

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Step 1.1.4.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{2}$ with respect to $x$ is $42 \frac{d}{d x} [x^{2}]$ .

$12 x^{3} - 60 x^{2} + 42 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36 x]$

Step 1.1.4.2

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

$12 x^{3} - 60 x^{2} + 42 (2 x) + \frac{d}{d x} [- 36 x]$

Step 1.1.4.3

Multiply $2$ by $42$ .

$12 x^{3} - 60 x^{2} + 84 x + \frac{d}{d x} [- 36 x]$

Step 1.1.5

Evaluate $\frac{d}{d x} [- 36 x]$ .

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Step 1.1.5.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x$ with respect to $x$ is $- 36 \frac{d}{d x} [x]$ .

$12 x^{3} - 60 x^{2} + 84 x - 36 \frac{d}{d x} [x]$

Step 1.1.5.2

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

$12 x^{3} - 60 x^{2} + 84 x - 36 \cdot 1$

Step 1.1.5.3

Multiply $- 36$ by $1$ .

$f' (x) = 12 x^{3} - 60 x^{2} + 84 x - 36$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{3} - 60 x^{2} + 84 x - 36$ .

$12 x^{3} - 60 x^{2} + 84 x - 36$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{3} - 60 x^{2} + 84 x - 36 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{3} - 60 x^{2} + 84 x - 36 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12$ out of $12 x^{3} - 60 x^{2} + 84 x - 36$ .

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Step 2.2.1.1

Factor $12$ out of $12 x^{3}$ .

$12 (x^{3}) - 60 x^{2} + 84 x - 36 = 0$

Step 2.2.1.2

Factor $12$ out of $- 60 x^{2}$ .

$12 (x^{3}) + 12 (- 5 x^{2}) + 84 x - 36 = 0$

Step 2.2.1.3

Factor $12$ out of $84 x$ .

$12 (x^{3}) + 12 (- 5 x^{2}) + 12 (7 x) - 36 = 0$

Step 2.2.1.4

Factor $12$ out of $- 36$ .

$12 x^{3} + 12 (- 5 x^{2}) + 12 (7 x) + 12 \cdot - 3 = 0$

Step 2.2.1.5

Factor $12$ out of $12 x^{3} + 12 (- 5 x^{2})$ .

$12 (x^{3} - 5 x^{2}) + 12 (7 x) + 12 \cdot - 3 = 0$

Step 2.2.1.6

Factor $12$ out of $12 (x^{3} - 5 x^{2}) + 12 (7 x)$ .

$12 (x^{3} - 5 x^{2} + 7 x) + 12 \cdot - 3 = 0$

Step 2.2.1.7

Factor $12$ out of $12 (x^{3} - 5 x^{2} + 7 x) + 12 \cdot - 3$ .

$12 (x^{3} - 5 x^{2} + 7 x - 3) = 0$

Step 2.2.2

Factor $x^{3} - 5 x^{2} + 7 x - 3$ using the rational roots test.

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Step 2.2.2.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1$

Step 2.2.2.2

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

$\pm 1, \pm 3$

Step 2.2.2.3

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 2.2.2.3.1

Substitute $1$ into the polynomial.

$1^{3} - 5 \cdot 1^{2} + 7 \cdot 1 - 3$

Step 2.2.2.3.2

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

$1 - 5 \cdot 1^{2} + 7 \cdot 1 - 3$

Step 2.2.2.3.3

Raise $1$ to the power of $2$ .

$1 - 5 \cdot 1 + 7 \cdot 1 - 3$

Step 2.2.2.3.4

Multiply $- 5$ by $1$ .

$1 - 5 + 7 \cdot 1 - 3$

Step 2.2.2.3.5

Subtract $5$ from $1$ .

$- 4 + 7 \cdot 1 - 3$

Step 2.2.2.3.6

Multiply $7$ by $1$ .

$- 4 + 7 - 3$

Step 2.2.2.3.7

Add $- 4$ and $7$ .

$3 - 3$

Step 2.2.2.3.8

Subtract $3$ from $3$ .

$0$

Step 2.2.2.4

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.

$\frac{x^{3} - 5 x^{2} + 7 x - 3}{x - 1}$

Step 2.2.2.5

Divide $x^{3} - 5 x^{2} + 7 x - 3$ by $x - 1$ .

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Step 2.2.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$5 x^{2}$

$7 x$

$3$

Step 2.2.2.5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$

Step 2.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$

Step 2.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - x^{2}$

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$

Step 2.2.2.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$

Step 2.2.2.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$

Step 2.2.2.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$

Step 2.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					-	$4 x^{2}$	+	$4 x$

Step 2.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 4 x$

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$

Step 2.2.2.5.10

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$

Step 2.2.2.5.11

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

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$	-	$3$

Step 2.2.2.5.12

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$	-	$3$

Step 2.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$	-	$3$
							+	$3 x$	-	$3$

Step 2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 3$

				$x^{2}$	-	$4 x$	+	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$	-	$3$
							-	$3 x$	+	$3$

Step 2.2.2.5.15

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

				$x^{2}$	-	$4 x$	+	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$7 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$7 x$
					+	$4 x^{2}$	-	$4 x$
							+	$3 x$	-	$3$
							-	$3 x$	+	$3$
										$0$

Step 2.2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 4 x + 3$

Step 2.2.2.6

Write $x^{3} - 5 x^{2} + 7 x - 3$ as a set of factors.

$12 ((x - 1) (x^{2} - 4 x + 3)) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.2.3.1.1

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.2.3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.2.3.1.1.2

Write the factored form using these integers.

$12 ((x - 1) ((x - 3) (x - 1))) = 0$

Step 2.2.3.1.2

Remove unnecessary parentheses.

$12 ((x - 1) (x - 3) (x - 1)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$12 (x - 1) (x - 3) (x - 1) = 0$

Step 2.2.4

Combine exponents.

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Step 2.2.4.1

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

$12 ((x - 1) (x - 1)) (x - 3) = 0$

Step 2.2.4.2

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

$12 ((x - 1) (x - 1)) (x - 3) = 0$

Step 2.2.4.3

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

$12 {(x - 1)}^{1 + 1} (x - 3) = 0$

Step 2.2.4.4

Add $1$ and $1$ .

$12 {(x - 1)}^{2} (x - 3) = 0$

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 $0$ .

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

$x - 3 = 0$

Step 2.4

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 2.4.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $12 {(x - 1)}^{2} (x - 3) = 0$ true.

$x = 1, 3$

Step 3

The values which make the derivative equal to $0$ are $1, 3$ .

$1, 3$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 1) \cup (1, 3) \cup (3, \infty)$

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.

$f' (0) = 12 {(0)}^{3} - 60 {(0)}^{2} + 84 (0) - 36$

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 $0$ .

$f' (0) = 12 \cdot 0 - 60 {(0)}^{2} + 84 (0) - 36$

Step 5.2.1.2

Multiply $12$ by $0$ .

$f' (0) = 0 - 60 {(0)}^{2} + 84 (0) - 36$

Step 5.2.1.3

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

$f' (0) = 0 - 60 \cdot 0 + 84 (0) - 36$

Step 5.2.1.4

Multiply $- 60$ by $0$ .

$f' (0) = 0 + 0 + 84 (0) - 36$

Step 5.2.1.5

Multiply $84$ by $0$ .

$f' (0) = 0 + 0 + 0 - 36$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 0 - 36$

Step 5.2.2.2

Add $0$ and $0$ .

$f' (0) = 0 - 36$

Step 5.2.2.3

Subtract $36$ from $0$ .

$f' (0) = - 36$

Step 5.2.3

The final answer is $- 36$ .

$- 36$

Step 5.3

At $x = 0$ the derivative is $- 36$ . 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 $(1, 3)$ 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.

$f' (2) = 12 {(2)}^{3} - 60 {(2)}^{2} + 84 (2) - 36$

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 $3$ .

$f' (2) = 12 \cdot 8 - 60 {(2)}^{2} + 84 (2) - 36$

Step 6.2.1.2

Multiply $12$ by $8$ .

$f' (2) = 96 - 60 {(2)}^{2} + 84 (2) - 36$

Step 6.2.1.3

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

$f' (2) = 96 - 60 \cdot 4 + 84 (2) - 36$

Step 6.2.1.4

Multiply $- 60$ by $4$ .

$f' (2) = 96 - 240 + 84 (2) - 36$

Step 6.2.1.5

Multiply $84$ by $2$ .

$f' (2) = 96 - 240 + 168 - 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $240$ from $96$ .

$f' (2) = - 144 + 168 - 36$

Step 6.2.2.2

Add $- 144$ and $168$ .

$f' (2) = 24 - 36$

Step 6.2.2.3

Subtract $36$ from $24$ .

$f' (2) = - 12$

Step 6.2.3

The final answer is $- 12$ .

$- 12$

Step 6.3

At $x = 2$ the derivative is $- 12$ . Since this is negative, the function is decreasing on $(1, 3)$ .

Decreasing on $(1, 3)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (4) = 12 {(4)}^{3} - 60 {(4)}^{2} + 84 (4) - 36$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $4$ to the power of $3$ .

$f' (4) = 12 \cdot 64 - 60 {(4)}^{2} + 84 (4) - 36$

Step 7.2.1.2

Multiply $12$ by $64$ .

$f' (4) = 768 - 60 {(4)}^{2} + 84 (4) - 36$

Step 7.2.1.3

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

$f' (4) = 768 - 60 \cdot 16 + 84 (4) - 36$

Step 7.2.1.4

Multiply $- 60$ by $16$ .

$f' (4) = 768 - 960 + 84 (4) - 36$

Step 7.2.1.5

Multiply $84$ by $4$ .

$f' (4) = 768 - 960 + 336 - 36$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $960$ from $768$ .

$f' (4) = - 192 + 336 - 36$

Step 7.2.2.2

Add $- 192$ and $336$ .

$f' (4) = 144 - 36$

Step 7.2.2.3

Subtract $36$ from $144$ .

$f' (4) = 108$

Step 7.2.3

The final answer is $108$ .

$108$

Step 7.3

At $x = 4$ the derivative is $108$ . Since this is positive, the function is increasing on $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(3, \infty)$

Decreasing on: $(- \infty, 1), (1, 3)$

Step 9