Calculus Examples

$\frac{9}{2} x^{4} - 8 x^{3} - 15 x^{2} + 12 x - 2$

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 $\frac{9}{2} x^{4} - 8 x^{3} - 15 x^{2} + 12 x - 2$ with respect to $x$ is $\frac{d}{d x} [\frac{9}{2} x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [\frac{9}{2} x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{9}{2} x^{4}]$ .

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Step 1.1.2.1

Since $\frac{9}{2}$ is constant with respect to $x$ , the derivative of $\frac{9}{2} x^{4}$ with respect to $x$ is $\frac{9}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{9}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

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 $n = 4$ .

$\frac{9}{2} (4 x^{3}) + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.3

Combine $4$ and $\frac{9}{2}$ .

$\frac{4 \cdot 9}{2} x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.4

Multiply $4$ by $9$ .

$\frac{36}{2} x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.5

Combine $\frac{36}{2}$ and $x^{3}$ .

$\frac{36 x^{3}}{2} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.6

Cancel the common factor of $36$ and $2$ .

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Step 1.1.2.6.1

Factor $2$ out of $36 x^{3}$ .

$\frac{2 (18 x^{3})}{2} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (18 x^{3})}{2 (1)} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{2 (18 x^{3})}{2 \cdot 1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{18 x^{3}}{1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.6.2.4

Divide $18 x^{3}$ by $1$ .

$18 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$18 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

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 $n = 3$ .

$18 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.3.3

Multiply $3$ by $- 8$ .

$18 x^{3} - 24 x^{2} + \frac{d}{d x} [- 15 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 15 x^{2}]$ .

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Step 1.1.4.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x^{2}$ with respect to $x$ is $- 15 \frac{d}{d x} [x^{2}]$ .

$18 x^{3} - 24 x^{2} - 15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

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 $n = 2$ .

$18 x^{3} - 24 x^{2} - 15 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.4.3

Multiply $2$ by $- 15$ .

$18 x^{3} - 24 x^{2} - 30 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$

Step 1.1.5

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

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Step 1.1.5.1

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

$18 x^{3} - 24 x^{2} - 30 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

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 $n = 1$ .

$18 x^{3} - 24 x^{2} - 30 x + 12 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.1.5.3

Multiply $12$ by $1$ .

$18 x^{3} - 24 x^{2} - 30 x + 12 + \frac{d}{d x} [- 2]$

Step 1.1.6

Differentiate using the Constant Rule.

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Step 1.1.6.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$18 x^{3} - 24 x^{2} - 30 x + 12 + 0$

Step 1.1.6.2

Add $18 x^{3} - 24 x^{2} - 30 x + 12$ and $0$ .

$f' (x) = 18 x^{3} - 24 x^{2} - 30 x + 12$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $18 x^{3} - 24 x^{2} - 30 x + 12$ .

$18 x^{3} - 24 x^{2} - 30 x + 12$

Step 2

Set the first derivative equal to $0$ then solve the equation $18 x^{3} - 24 x^{2} - 30 x + 12 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$18 x^{3} - 24 x^{2} - 30 x + 12 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $6$ out of $18 x^{3} - 24 x^{2} - 30 x + 12$ .

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Step 2.2.1.1

Factor $6$ out of $18 x^{3}$ .

$6 (3 x^{3}) - 24 x^{2} - 30 x + 12 = 0$

Step 2.2.1.2

Factor $6$ out of $- 24 x^{2}$ .

$6 (3 x^{3}) + 6 (- 4 x^{2}) - 30 x + 12 = 0$

Step 2.2.1.3

Factor $6$ out of $- 30 x$ .

$6 (3 x^{3}) + 6 (- 4 x^{2}) + 6 (- 5 x) + 12 = 0$

Step 2.2.1.4

Factor $6$ out of $12$ .

$6 (3 x^{3}) + 6 (- 4 x^{2}) + 6 (- 5 x) + 6 (2) = 0$

Step 2.2.1.5

Factor $6$ out of $6 (3 x^{3}) + 6 (- 4 x^{2})$ .

$6 (3 x^{3} - 4 x^{2}) + 6 (- 5 x) + 6 (2) = 0$

Step 2.2.1.6

Factor $6$ out of $6 (3 x^{3} - 4 x^{2}) + 6 (- 5 x)$ .

$6 (3 x^{3} - 4 x^{2} - 5 x) + 6 (2) = 0$

Step 2.2.1.7

Factor $6$ out of $6 (3 x^{3} - 4 x^{2} - 5 x) + 6 (2)$ .

$6 (3 x^{3} - 4 x^{2} - 5 x + 2) = 0$

Step 2.2.2

Factor $3 x^{3} - 4 x^{2} - 5 x + 2$ 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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

Step 2.2.2.2

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

$\pm 1, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

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.

$3 {(- 1)}^{3} - 4 {(- 1)}^{2} - 5 \cdot - 1 + 2$

Step 2.2.2.3.2

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

$3 \cdot - 1 - 4 {(- 1)}^{2} - 5 \cdot - 1 + 2$

Step 2.2.2.3.3

Multiply $3$ by $- 1$ .

$- 3 - 4 {(- 1)}^{2} - 5 \cdot - 1 + 2$

Step 2.2.2.3.4

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

$- 3 - 4 \cdot 1 - 5 \cdot - 1 + 2$

Step 2.2.2.3.5

Multiply $- 4$ by $1$ .

$- 3 - 4 - 5 \cdot - 1 + 2$

Step 2.2.2.3.6

Subtract $4$ from $- 3$ .

$- 7 - 5 \cdot - 1 + 2$

Step 2.2.2.3.7

Multiply $- 5$ by $- 1$ .

$- 7 + 5 + 2$

Step 2.2.2.3.8

Add $- 7$ and $5$ .

$- 2 + 2$

Step 2.2.2.3.9

Add $- 2$ and $2$ .

$0$

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.

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

Step 2.2.2.5

Divide $3 x^{3} - 4 x^{2} - 5 x + 2$ 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 $0$ .

$x$

$1$

$3 x^{3}$

$4 x^{2}$

$5 x$

$2$

Step 2.2.2.5.2

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

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

Step 2.2.2.5.3

Multiply the new quotient term by the divisor.

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

Step 2.2.2.5.4

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

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

Step 2.2.2.5.5

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

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

Step 2.2.2.5.6

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

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

Step 2.2.2.5.7

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

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

Step 2.2.2.5.8

Multiply the new quotient term by the divisor.

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

Step 2.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} - 7 x$

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

Step 2.2.2.5.10

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

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

Step 2.2.2.5.11

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

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

Step 2.2.2.5.12

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

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

Step 2.2.2.5.13

Multiply the new quotient term by the divisor.

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

Step 2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 2$

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

Step 2.2.2.5.15

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

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

Step 2.2.2.5.16

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

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

Step 2.2.2.6

Write $3 x^{3} - 4 x^{2} - 5 x + 2$ as a set of factors.

$6 ((x + 1) (3 x^{2} - 7 x + 2)) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor by grouping.

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Step 2.2.3.1.1

Factor by grouping.

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Step 2.2.3.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 7$ .

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Step 2.2.3.1.1.1.1

Factor $- 7$ out of $- 7 x$ .

$6 ((x + 1) (3 x^{2} - 7 x + 2)) = 0$

Step 2.2.3.1.1.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$6 ((x + 1) (3 x^{2} + (- 1 - 6) x + 2)) = 0$

Step 2.2.3.1.1.1.3

Apply the distributive property.

$6 ((x + 1) (3 x^{2} - 1 x - 6 x + 2)) = 0$

Step 2.2.3.1.1.2

Factor out the greatest common factor from each group.

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Step 2.2.3.1.1.2.1

Group the first two terms and the last two terms.

$6 ((x + 1) ((3 x^{2} - 1 x) - 6 x + 2)) = 0$

Step 2.2.3.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 ((x + 1) (x (3 x - 1) - 2 (3 x - 1))) = 0$

Step 2.2.3.1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$6 ((x + 1) ((3 x - 1) (x - 2))) = 0$

Step 2.2.3.1.2

Remove unnecessary parentheses.

$6 ((x + 1) (3 x - 1) (x - 2)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$6 (x + 1) (3 x - 1) (x - 2) = 0$

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 $0$ .

$x + 1 = 0$

$3 x - 1 = 0$

$x - 2 = 0$

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

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 2.5.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 2.5.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 2.5.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 2.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.7

The final solution is all the values that make $6 (x + 1) (3 x - 1) (x - 2) = 0$ true.

$x = - 1, \frac{1}{3}, 2$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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 4

Evaluate $\frac{9}{2} x^{4} - 8 x^{3} - 15 x^{2} + 12 x - 2$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

$\frac{9}{2} \cdot {(- 1)}^{4} - 8 {(- 1)}^{3} - 15 {(- 1)}^{2} + 12 (- 1) - 2$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $- 1$ to the power of $4$ .

$\frac{9}{2} \cdot 1 - 8 {(- 1)}^{3} - 15 {(- 1)}^{2} + 12 (- 1) - 2$

Step 4.1.2.1.2

Multiply $\frac{9}{2}$ by $1$ .

$\frac{9}{2} - 8 {(- 1)}^{3} - 15 {(- 1)}^{2} + 12 (- 1) - 2$

Step 4.1.2.1.3

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

$\frac{9}{2} - 8 \cdot - 1 - 15 {(- 1)}^{2} + 12 (- 1) - 2$

Step 4.1.2.1.4

Multiply $- 8$ by $- 1$ .

$\frac{9}{2} + 8 - 15 {(- 1)}^{2} + 12 (- 1) - 2$

Step 4.1.2.1.5

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

$\frac{9}{2} + 8 - 15 \cdot 1 + 12 (- 1) - 2$

Step 4.1.2.1.6

Multiply $- 15$ by $1$ .

$\frac{9}{2} + 8 - 15 + 12 (- 1) - 2$

Step 4.1.2.1.7

Multiply $12$ by $- 1$ .

$\frac{9}{2} + 8 - 15 - 12 - 2$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $8$ as a fraction with denominator $1$ .

$\frac{9}{2} + \frac{8}{1} - 15 - 12 - 2$

Step 4.1.2.2.2

Multiply $\frac{8}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8}{1} \cdot \frac{2}{2} - 15 - 12 - 2$

Step 4.1.2.2.3

Multiply $\frac{8}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} - 15 - 12 - 2$

Step 4.1.2.2.4

Write $- 15$ as a fraction with denominator $1$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15}{1} - 12 - 2$

Step 4.1.2.2.5

Multiply $\frac{- 15}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15}{1} \cdot \frac{2}{2} - 12 - 2$

Step 4.1.2.2.6

Multiply $\frac{- 15}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} - 12 - 2$

Step 4.1.2.2.7

Write $- 12$ as a fraction with denominator $1$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12}{1} - 2$

Step 4.1.2.2.8

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12}{1} \cdot \frac{2}{2} - 2$

Step 4.1.2.2.9

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12 \cdot 2}{2} - 2$

Step 4.1.2.2.10

Write $- 2$ as a fraction with denominator $1$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12 \cdot 2}{2} + \frac{- 2}{1}$

Step 4.1.2.2.11

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2}$

Step 4.1.2.2.12

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$\frac{9}{2} + \frac{8 \cdot 2}{2} + \frac{- 15 \cdot 2}{2} + \frac{- 12 \cdot 2}{2} + \frac{- 2 \cdot 2}{2}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{9 + 8 \cdot 2 - 15 \cdot 2 - 12 \cdot 2 - 2 \cdot 2}{2}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Multiply $8$ by $2$ .

$\frac{9 + 16 - 15 \cdot 2 - 12 \cdot 2 - 2 \cdot 2}{2}$

Step 4.1.2.4.2

Multiply $- 15$ by $2$ .

$\frac{9 + 16 - 30 - 12 \cdot 2 - 2 \cdot 2}{2}$

Step 4.1.2.4.3

Multiply $- 12$ by $2$ .

$\frac{9 + 16 - 30 - 24 - 2 \cdot 2}{2}$

Step 4.1.2.4.4

Multiply $- 2$ by $2$ .

$\frac{9 + 16 - 30 - 24 - 4}{2}$

Step 4.1.2.5

Simplify the expression.

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Step 4.1.2.5.1

Add $9$ and $16$ .

$\frac{25 - 30 - 24 - 4}{2}$

Step 4.1.2.5.2

Subtract $30$ from $25$ .

$\frac{- 5 - 24 - 4}{2}$

Step 4.1.2.5.3

Subtract $24$ from $- 5$ .

$\frac{- 29 - 4}{2}$

Step 4.1.2.5.4

Subtract $4$ from $- 29$ .

$\frac{- 33}{2}$

Step 4.1.2.5.5

Move the negative in front of the fraction.

$- \frac{33}{2}$

Step 4.2

Evaluate at $x = \frac{1}{3}$ .

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Step 4.2.1

Substitute $\frac{1}{3}$ for $x$ .

$\frac{9}{2} \cdot {(\frac{1}{3})}^{4} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{9}{2} \cdot \frac{1^{4}}{3^{4}} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.2

Combine.

$\frac{9 \cdot 1^{4}}{2 \cdot 3^{4}} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.3

One to any power is one.

$\frac{9 \cdot 1}{2 \cdot 3^{4}} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.4

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

$\frac{9 \cdot 1}{2 \cdot 81} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.5

Multiply $9$ by $1$ .

$\frac{9}{2 \cdot 81} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.6

Multiply $2$ by $81$ .

$\frac{9}{162} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.7

Cancel the common factor of $9$ and $162$ .

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Step 4.2.2.1.7.1

Factor $9$ out of $9$ .

$\frac{9 (1)}{162} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.7.2

Cancel the common factors.

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Step 4.2.2.1.7.2.1

Factor $9$ out of $162$ .

$\frac{9 \cdot 1}{9 \cdot 18} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.7.2.2

Cancel the common factor.

$\frac{9 \cdot 1}{9 \cdot 18} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.7.2.3

Rewrite the expression.

$\frac{1}{18} - 8 {(\frac{1}{3})}^{3} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.8

Apply the product rule to $\frac{1}{3}$ .

$\frac{1}{18} - 8 \frac{1^{3}}{3^{3}} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.9

One to any power is one.

$\frac{1}{18} - 8 \frac{1}{3^{3}} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.10

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

$\frac{1}{18} - 8 (\frac{1}{27}) - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.11

Combine $- 8$ and $\frac{1}{27}$ .

$\frac{1}{18} + \frac{- 8}{27} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.12

Move the negative in front of the fraction.

$\frac{1}{18} - \frac{8}{27} - 15 {(\frac{1}{3})}^{2} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.13

Apply the product rule to $\frac{1}{3}$ .

$\frac{1}{18} - \frac{8}{27} - 15 \frac{1^{2}}{3^{2}} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.14

One to any power is one.

$\frac{1}{18} - \frac{8}{27} - 15 \frac{1}{3^{2}} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.15

Raise $3$ to the power of $2$ .

$\frac{1}{18} - \frac{8}{27} - 15 (\frac{1}{9}) + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.16

Cancel the common factor of $3$ .

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Step 4.2.2.1.16.1

Factor $3$ out of $- 15$ .

$\frac{1}{18} - \frac{8}{27} + 3 (- 5) \frac{1}{9} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.16.2

Factor $3$ out of $9$ .

$\frac{1}{18} - \frac{8}{27} + 3 \cdot - 5 \frac{1}{3 \cdot 3} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.16.3

Cancel the common factor.

$\frac{1}{18} - \frac{8}{27} + 3 \cdot - 5 \frac{1}{3 \cdot 3} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.16.4

Rewrite the expression.

$\frac{1}{18} - \frac{8}{27} - 5 (\frac{1}{3}) + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.17

Combine $- 5$ and $\frac{1}{3}$ .

$\frac{1}{18} - \frac{8}{27} + \frac{- 5}{3} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.18

Move the negative in front of the fraction.

$\frac{1}{18} - \frac{8}{27} - \frac{5}{3} + 12 (\frac{1}{3}) - 2$

Step 4.2.2.1.19

Cancel the common factor of $3$ .

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Step 4.2.2.1.19.1

Factor $3$ out of $12$ .

$\frac{1}{18} - \frac{8}{27} - \frac{5}{3} + 3 (4) \frac{1}{3} - 2$

Step 4.2.2.1.19.2

Cancel the common factor.

$\frac{1}{18} - \frac{8}{27} - \frac{5}{3} + 3 \cdot 4 \frac{1}{3} - 2$

Step 4.2.2.1.19.3

Rewrite the expression.

$\frac{1}{18} - \frac{8}{27} - \frac{5}{3} + 4 - 2$

Step 4.2.2.2

Find the common denominator.

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Step 4.2.2.2.1

Multiply $\frac{1}{18}$ by $\frac{3}{3}$ .

$\frac{1}{18} \cdot \frac{3}{3} - \frac{8}{27} - \frac{5}{3} + 4 - 2$

Step 4.2.2.2.2

Multiply $\frac{1}{18}$ by $\frac{3}{3}$ .

$\frac{3}{18 \cdot 3} - \frac{8}{27} - \frac{5}{3} + 4 - 2$

Step 4.2.2.2.3

Multiply $\frac{8}{27}$ by $\frac{2}{2}$ .

$\frac{3}{18 \cdot 3} - (\frac{8}{27} \cdot \frac{2}{2}) - \frac{5}{3} + 4 - 2$

Step 4.2.2.2.4

Multiply $\frac{8}{27}$ by $\frac{2}{2}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5}{3} + 4 - 2$

Step 4.2.2.2.5

Multiply $\frac{5}{3}$ by $\frac{18}{18}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - (\frac{5}{3} \cdot \frac{18}{18}) + 4 - 2$

Step 4.2.2.2.6

Multiply $\frac{5}{3}$ by $\frac{18}{18}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + 4 - 2$

Step 4.2.2.2.7

Write $4$ as a fraction with denominator $1$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4}{1} - 2$

Step 4.2.2.2.8

Multiply $\frac{4}{1}$ by $\frac{54}{54}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4}{1} \cdot \frac{54}{54} - 2$

Step 4.2.2.2.9

Multiply $\frac{4}{1}$ by $\frac{54}{54}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} - 2$

Step 4.2.2.2.10

Write $- 2$ as a fraction with denominator $1$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2}{1}$

Step 4.2.2.2.11

Multiply $\frac{- 2}{1}$ by $\frac{54}{54}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2}{1} \cdot \frac{54}{54}$

Step 4.2.2.2.12

Multiply $\frac{- 2}{1}$ by $\frac{54}{54}$ .

$\frac{3}{18 \cdot 3} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.2.13

Reorder the factors of $18 \cdot 3$ .

$\frac{3}{3 \cdot 18} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.2.14

Multiply $3$ by $18$ .

$\frac{3}{54} - \frac{8 \cdot 2}{27 \cdot 2} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.2.15

Reorder the factors of $27 \cdot 2$ .

$\frac{3}{54} - \frac{8 \cdot 2}{2 \cdot 27} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.2.16

Multiply $2$ by $27$ .

$\frac{3}{54} - \frac{8 \cdot 2}{54} - \frac{5 \cdot 18}{3 \cdot 18} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.2.17

Multiply $3$ by $18$ .

$\frac{3}{54} - \frac{8 \cdot 2}{54} - \frac{5 \cdot 18}{54} + \frac{4 \cdot 54}{54} + \frac{- 2 \cdot 54}{54}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$\frac{3 - 8 \cdot 2 - 5 \cdot 18 + 4 \cdot 54 - 2 \cdot 54}{54}$

Step 4.2.2.4

Simplify each term.

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Step 4.2.2.4.1

Multiply $- 8$ by $2$ .

$\frac{3 - 16 - 5 \cdot 18 + 4 \cdot 54 - 2 \cdot 54}{54}$

Step 4.2.2.4.2

Multiply $- 5$ by $18$ .

$\frac{3 - 16 - 90 + 4 \cdot 54 - 2 \cdot 54}{54}$

Step 4.2.2.4.3

Multiply $4$ by $54$ .

$\frac{3 - 16 - 90 + 216 - 2 \cdot 54}{54}$

Step 4.2.2.4.4

Multiply $- 2$ by $54$ .

$\frac{3 - 16 - 90 + 216 - 108}{54}$

Step 4.2.2.5

Simplify by adding and subtracting.

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Step 4.2.2.5.1

Subtract $16$ from $3$ .

$\frac{- 13 - 90 + 216 - 108}{54}$

Step 4.2.2.5.2

Subtract $90$ from $- 13$ .

$\frac{- 103 + 216 - 108}{54}$

Step 4.2.2.5.3

Add $- 103$ and $216$ .

$\frac{113 - 108}{54}$

Step 4.2.2.5.4

Subtract $108$ from $113$ .

$\frac{5}{54}$

Step 4.3

Evaluate at $x = 2$ .

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Step 4.3.1

Substitute $2$ for $x$ .

$\frac{9}{2} \cdot {(2)}^{4} - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.1

Factor $2$ out of ${(2)}^{4}$ .

$\frac{9}{2} \cdot (2 \cdot 2^{3}) - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.1.2

Cancel the common factor.

$\frac{9}{2} \cdot (2 \cdot 2^{3}) - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.1.3

Rewrite the expression.

$9 \cdot 2^{3} - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.2

Raise $2$ to the power of $3$ .

$9 \cdot 8 - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.3

Multiply $9$ by $8$ .

$72 - 8 {(2)}^{3} - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.4

Raise $2$ to the power of $3$ .

$72 - 8 \cdot 8 - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.5

Multiply $- 8$ by $8$ .

$72 - 64 - 15 {(2)}^{2} + 12 (2) - 2$

Step 4.3.2.1.6

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

$72 - 64 - 15 \cdot 4 + 12 (2) - 2$

Step 4.3.2.1.7

Multiply $- 15$ by $4$ .

$72 - 64 - 60 + 12 (2) - 2$

Step 4.3.2.1.8

Multiply $12$ by $2$ .

$72 - 64 - 60 + 24 - 2$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Subtract $64$ from $72$ .

$8 - 60 + 24 - 2$

Step 4.3.2.2.2

Subtract $60$ from $8$ .

$- 52 + 24 - 2$

Step 4.3.2.2.3

Add $- 52$ and $24$ .

$- 28 - 2$

Step 4.3.2.2.4

Subtract $2$ from $- 28$ .

$- 30$

Step 4.4

List all of the points.

$(- 1, - \frac{33}{2}), (\frac{1}{3}, \frac{5}{54}), (2, - 30)$

Step 5