Algebra Examples

$y = x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36$

Step 1

Write $y = x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36$ as a function.

$f (x) = x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.1.2

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

$4 x^{3} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.2

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

Tap for more steps...

Step 2.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.2.2

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

$4 x^{3} - 2 (3 x^{2}) + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.2.3

Multiply $3$ by $- 2$ .

$4 x^{3} - 6 x^{2} + \frac{d}{d x} [- 11 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.3

Evaluate $\frac{d}{d x} [- 11 x^{2}]$ .

Tap for more steps...

Step 2.3.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x^{2}$ with respect to $x$ is $- 11 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 6 x^{2} - 11 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.3.2

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

$4 x^{3} - 6 x^{2} - 11 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.3.3

Multiply $2$ by $- 11$ .

$4 x^{3} - 6 x^{2} - 22 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [36]$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$4 x^{3} - 6 x^{2} - 22 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [36]$

Step 2.4.2

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

$4 x^{3} - 6 x^{2} - 22 x + 12 \cdot 1 + \frac{d}{d x} [36]$

Step 2.4.3

Multiply $12$ by $1$ .

$4 x^{3} - 6 x^{2} - 22 x + 12 + \frac{d}{d x} [36]$

Step 2.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.5.1

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$4 x^{3} - 6 x^{2} - 22 x + 12 + 0$

Step 2.5.2

Add $4 x^{3} - 6 x^{2} - 22 x + 12$ and $0$ .

$4 x^{3} - 6 x^{2} - 22 x + 12$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $4 x^{3} - 6 x^{2} - 22 x + 12$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 22 x] + \frac{d}{d x} [12]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.2.2

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

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.3

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

Tap for more steps...

Step 3.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.3.2

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

$f'' (x) = 12 x^{2} - 6 (2 x) + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.3.3

Multiply $2$ by $- 6$ .

$f'' (x) = 12 x^{2} - 12 x + \frac{d}{d x} (- 22 x) + \frac{d}{d x} (12)$

Step 3.4

Evaluate $\frac{d}{d x} [- 22 x]$ .

Tap for more steps...

Step 3.4.1

Since $- 22$ is constant with respect to $x$ , the derivative of $- 22 x$ with respect to $x$ is $- 22 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} - 12 x - 22 \frac{d}{d x} x + \frac{d}{d x} (12)$

Step 3.4.2

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

$f'' (x) = 12 x^{2} - 12 x - 22 \cdot 1 + \frac{d}{d x} (12)$

Step 3.4.3

Multiply $- 22$ by $1$ .

$f'' (x) = 12 x^{2} - 12 x - 22 + \frac{d}{d x} (12)$

Step 3.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f'' (x) = 12 x^{2} - 12 x - 22 + 0$

Step 3.5.2

Add $12 x^{2} - 12 x - 22$ and $0$ .

$f'' (x) = 12 x^{2} - 12 x - 22$

Step 4

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

$4 x^{3} - 6 x^{2} - 22 x + 12 = 0$

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

Differentiate.

Tap for more steps...

Step 5.1.1.1

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 2 x^{3}) + \frac{d}{d x} (- 11 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.1.2

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

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 2 x^{3}) + \frac{d}{d x} (- 11 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.2

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

Tap for more steps...

Step 5.1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 x^{3} - 2 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 11 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

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 $n = 3$ .

$f' (x) = 4 x^{3} - 2 (3 x^{2}) + \frac{d}{d x} (- 11 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.2.3

Multiply $3$ by $- 2$ .

$f' (x) = 4 x^{3} - 6 x^{2} + \frac{d}{d x} (- 11 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 11 x^{2}]$ .

Tap for more steps...

Step 5.1.3.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x^{2}$ with respect to $x$ is $- 11 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 11 \frac{d}{d x} x^{2} + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

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 $n = 2$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 11 (2 x) + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.3.3

Multiply $2$ by $- 11$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + \frac{d}{d x} (12 x) + \frac{d}{d x} (36)$

Step 5.1.4

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

Tap for more steps...

Step 5.1.4.1

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

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + 12 \frac{d}{d x} (x) + \frac{d}{d x} (36)$

Step 5.1.4.2

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

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + 12 \cdot 1 + \frac{d}{d x} (36)$

Step 5.1.4.3

Multiply $12$ by $1$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + 12 + \frac{d}{d x} (36)$

Step 5.1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 5.1.5.1

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + 12 + 0$

Step 5.1.5.2

Add $4 x^{3} - 6 x^{2} - 22 x + 12$ and $0$ .

$f' (x) = 4 x^{3} - 6 x^{2} - 22 x + 12$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 6 x^{2} - 22 x + 12$ .

$4 x^{3} - 6 x^{2} - 22 x + 12$

Step 6

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 6 x^{2} - 22 x + 12 = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

$4 x^{3} - 6 x^{2} - 22 x + 12 = 0$

Step 6.2

Factor the left side of the equation.

Tap for more steps...

Step 6.2.1

Factor $2$ out of $4 x^{3} - 6 x^{2} - 22 x + 12$ .

Tap for more steps...

Step 6.2.1.1

Factor $2$ out of $4 x^{3}$ .

$2 (2 x^{3}) - 6 x^{2} - 22 x + 12 = 0$

Step 6.2.1.2

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

$2 (2 x^{3}) + 2 (- 3 x^{2}) - 22 x + 12 = 0$

Step 6.2.1.3

Factor $2$ out of $- 22 x$ .

$2 (2 x^{3}) + 2 (- 3 x^{2}) + 2 (- 11 x) + 12 = 0$

Step 6.2.1.4

Factor $2$ out of $12$ .

$2 (2 x^{3}) + 2 (- 3 x^{2}) + 2 (- 11 x) + 2 (6) = 0$

Step 6.2.1.5

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

$2 (2 x^{3} - 3 x^{2}) + 2 (- 11 x) + 2 (6) = 0$

Step 6.2.1.6

Factor $2$ out of $2 (2 x^{3} - 3 x^{2}) + 2 (- 11 x)$ .

$2 (2 x^{3} - 3 x^{2} - 11 x) + 2 (6) = 0$

Step 6.2.1.7

Factor $2$ out of $2 (2 x^{3} - 3 x^{2} - 11 x) + 2 (6)$ .

$2 (2 x^{3} - 3 x^{2} - 11 x + 6) = 0$

Step 6.2.2

Factor $2 x^{3} - 3 x^{2} - 11 x + 6$ using the rational roots test.

Tap for more steps...

Step 6.2.2.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1, \pm 2$

Step 6.2.2.2

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

$\pm 1, \pm 0.5, \pm 6, \pm 3, \pm 2, \pm 1.5$

Step 6.2.2.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Tap for more steps...

Step 6.2.2.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 6.2.2.3.2

Raise $0.5$ to the power of $3$ .

$2 \cdot 0.125 - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 6.2.2.3.3

Multiply $2$ by $0.125$ .

$0.25 - 3 \cdot {0.5}^{2} - 11 \cdot 0.5 + 6$

Step 6.2.2.3.4

Raise $0.5$ to the power of $2$ .

$0.25 - 3 \cdot 0.25 - 11 \cdot 0.5 + 6$

Step 6.2.2.3.5

Multiply $- 3$ by $0.25$ .

$0.25 - 0.75 - 11 \cdot 0.5 + 6$

Step 6.2.2.3.6

Subtract $0.75$ from $0.25$ .

$- 0.5 - 11 \cdot 0.5 + 6$

Step 6.2.2.3.7

Multiply $- 11$ by $0.5$ .

$- 0.5 - 5.5 + 6$

Step 6.2.2.3.8

Subtract $5.5$ from $- 0.5$ .

$- 6 + 6$

Step 6.2.2.3.9

Add $- 6$ and $6$ .

$0$

Step 6.2.2.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 3 x^{2} - 11 x + 6}{2 x - 1}$

Step 6.2.2.5

Divide $2 x^{3} - 3 x^{2} - 11 x + 6$ by $2 x - 1$ .

Tap for more steps...

Step 6.2.2.5.1

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

$2 x$

$1$

$2 x^{3}$

$3 x^{2}$

$11 x$

$6$

Step 6.2.2.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

					$x^{2}$
	$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$

Step 6.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			+	$2 x^{3}$	-	$x^{2}$

Step 6.2.2.5.4

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$

Step 6.2.2.5.5

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$

Step 6.2.2.5.6

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 6.2.2.5.7

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 6.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					-	$2 x^{2}$	+	$x$

Step 6.2.2.5.9

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$

Step 6.2.2.5.10

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$

Step 6.2.2.5.11

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

				$x^{2}$	-	$x$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$

Step 6.2.2.5.12

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$

Step 6.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							-	$12 x$	+	$6$

Step 6.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 6$

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							+	$12 x$	-	$6$

Step 6.2.2.5.15

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

				$x^{2}$	-	$x$	-	$6$
$2 x$	-	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$11 x$	+	$6$
			-	$2 x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$x$
							-	$12 x$	+	$6$
							+	$12 x$	-	$6$
										$0$

Step 6.2.2.5.16

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

$x^{2} - x - 6$

Step 6.2.2.6

Write $2 x^{3} - 3 x^{2} - 11 x + 6$ as a set of factors.

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

Step 6.2.3

Factor.

Tap for more steps...

Step 6.2.3.1

Factor $x^{2} - x - 6$ using the AC method.

Tap for more steps...

Step 6.2.3.1.1

Factor $x^{2} - x - 6$ using the AC method.

Tap for more steps...

Step 6.2.3.1.1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 6.2.3.1.1.2

Write the factored form using these integers.

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

Step 6.2.3.1.2

Remove unnecessary parentheses.

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

Step 6.2.3.2

Remove unnecessary parentheses.

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

Step 6.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 1 = 0$

$x - 3 = 0$

$x + 2 = 0$

Step 6.4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.4.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 6.4.2

Solve $2 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 6.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 6.4.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

Tap for more steps...

Step 6.4.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6.5

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

Tap for more steps...

Step 6.5.1

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

$x - 3 = 0$

Step 6.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.6

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

Tap for more steps...

Step 6.6.1

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

$x + 2 = 0$

Step 6.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.7

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

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

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.

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

Step 9

Evaluate the second derivative at $x = \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(\frac{1}{2})}^{2} - 12 (\frac{1}{2}) - 22$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Apply the product rule to $\frac{1}{2}$ .

$12 \frac{1^{2}}{2^{2}} - 12 (\frac{1}{2}) - 22$

Step 10.1.2

One to any power is one.

$12 \frac{1}{2^{2}} - 12 (\frac{1}{2}) - 22$

Step 10.1.3

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

$12 (\frac{1}{4}) - 12 (\frac{1}{2}) - 22$

Step 10.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 10.1.4.1

Factor $4$ out of $12$ .

$4 (3) \frac{1}{4} - 12 (\frac{1}{2}) - 22$

Step 10.1.4.2

Cancel the common factor.

$4 \cdot 3 \frac{1}{4} - 12 (\frac{1}{2}) - 22$

Step 10.1.4.3

Rewrite the expression.

$3 - 12 (\frac{1}{2}) - 22$

Step 10.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1.5.1

Factor $2$ out of $- 12$ .

$3 + 2 (- 6) \frac{1}{2} - 22$

Step 10.1.5.2

Cancel the common factor.

$3 + 2 \cdot - 6 \frac{1}{2} - 22$

Step 10.1.5.3

Rewrite the expression.

$3 - 6 - 22$

Step 10.2

Simplify by subtracting numbers.

Tap for more steps...

Step 10.2.1

Subtract $6$ from $3$ .

$- 3 - 22$

Step 10.2.2

Subtract $22$ from $- 3$ .

$- 25$

Step 11

$x = \frac{1}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{1}{2}$ is a local maximum

Step 12

Find the y-value when $x = \frac{1}{2}$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = {(\frac{1}{2})}^{4} - 2 {(\frac{1}{2})}^{3} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

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

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1^{4}}{2^{4}} - 2 {(\frac{1}{2})}^{3} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.2

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{2^{4}} - 2 {(\frac{1}{2})}^{3} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.3

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

$f (\frac{1}{2}) = \frac{1}{16} - 2 {(\frac{1}{2})}^{3} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.4

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{16} - 2 \frac{1^{3}}{2^{3}} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.5

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{16} - 2 \frac{1}{2^{3}} - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.6

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

$f (\frac{1}{2}) = \frac{1}{16} - 2 (\frac{1}{8}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.2.1.7.1

Factor $2$ out of $- 2$ .

$f (\frac{1}{2}) = \frac{1}{16} + 2 (- 1) (\frac{1}{8}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.7.2

Factor $2$ out of $8$ .

$f (\frac{1}{2}) = \frac{1}{16} + 2 \cdot (- 1 \frac{1}{2 \cdot 4}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.7.3

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{16} + 2 \cdot (- 1 \frac{1}{2 \cdot 4}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.7.4

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{16} - 1 (\frac{1}{4}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.8

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$f (\frac{1}{2}) = \frac{1}{16} - (\frac{1}{4}) - 11 {(\frac{1}{2})}^{2} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.9

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - 11 \frac{1^{2}}{2^{2}} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.10

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - 11 \frac{1}{2^{2}} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.11

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

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - 11 (\frac{1}{4}) + 12 (\frac{1}{2}) + 36$

Step 12.2.1.12

Combine $- 11$ and $\frac{1}{4}$ .

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} + \frac{- 11}{4} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.13

Move the negative in front of the fraction.

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - \frac{11}{4} + 12 (\frac{1}{2}) + 36$

Step 12.2.1.14

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.2.1.14.1

Factor $2$ out of $12$ .

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - \frac{11}{4} + 2 (6) (\frac{1}{2}) + 36$

Step 12.2.1.14.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - \frac{11}{4} + 2 \cdot (6 (\frac{1}{2})) + 36$

Step 12.2.1.14.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} - \frac{11}{4} + 6 + 36$

Step 12.2.2

Combine fractions.

Tap for more steps...

Step 12.2.2.1

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = 6 + 36 + \frac{1}{16} + \frac{- 1 - 11}{4}$

Step 12.2.2.2

Subtract $11$ from $- 1$ .

$f (\frac{1}{2}) = 6 + 36 + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3

Find the common denominator.

Tap for more steps...

Step 12.2.3.1

Write $6$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{6}{1} + 36 + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.2

Multiply $\frac{6}{1}$ by $\frac{16}{16}$ .

$f (\frac{1}{2}) = \frac{6}{1} \cdot \frac{16}{16} + 36 + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.3

Multiply $\frac{6}{1}$ by $\frac{16}{16}$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + 36 + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.4

Write $36$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36}{1} + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.5

Multiply $\frac{36}{1}$ by $\frac{16}{16}$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36}{1} \cdot \frac{16}{16} + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.6

Multiply $\frac{36}{1}$ by $\frac{16}{16}$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36 \cdot 16}{16} + \frac{1}{16} + \frac{- 12}{4}$

Step 12.2.3.7

Multiply $\frac{- 12}{4}$ by $\frac{4}{4}$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36 \cdot 16}{16} + \frac{1}{16} + \frac{- 12}{4} \cdot \frac{4}{4}$

Step 12.2.3.8

Multiply $\frac{- 12}{4}$ by $\frac{4}{4}$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36 \cdot 16}{16} + \frac{1}{16} + \frac{- 12 \cdot 4}{4 \cdot 4}$

Step 12.2.3.9

Multiply $4$ by $4$ .

$f (\frac{1}{2}) = \frac{6 \cdot 16}{16} + \frac{36 \cdot 16}{16} + \frac{1}{16} + \frac{- 12 \cdot 4}{16}$

Step 12.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{6 \cdot 16 + 36 \cdot 16 + 1 - 12 \cdot 4}{16}$

Step 12.2.5

Simplify each term.

Tap for more steps...

Step 12.2.5.1

Multiply $6$ by $16$ .

$f (\frac{1}{2}) = \frac{96 + 36 \cdot 16 + 1 - 12 \cdot 4}{16}$

Step 12.2.5.2

Multiply $36$ by $16$ .

$f (\frac{1}{2}) = \frac{96 + 576 + 1 - 12 \cdot 4}{16}$

Step 12.2.5.3

Multiply $- 12$ by $4$ .

$f (\frac{1}{2}) = \frac{96 + 576 + 1 - 48}{16}$

Step 12.2.6

Simplify by adding and subtracting.

Tap for more steps...

Step 12.2.6.1

Add $96$ and $576$ .

$f (\frac{1}{2}) = \frac{672 + 1 - 48}{16}$

Step 12.2.6.2

Add $672$ and $1$ .

$f (\frac{1}{2}) = \frac{673 - 48}{16}$

Step 12.2.6.3

Subtract $48$ from $673$ .

$f (\frac{1}{2}) = \frac{625}{16}$

Step 12.2.7

The final answer is $\frac{625}{16}$ .

$y = \frac{625}{16}$

Step 13

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(3)}^{2} - 12 \cdot 3 - 22$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Simplify each term.

Tap for more steps...

Step 14.1.1

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

$12 \cdot 9 - 12 \cdot 3 - 22$

Step 14.1.2

Multiply $12$ by $9$ .

$108 - 12 \cdot 3 - 22$

Step 14.1.3

Multiply $- 12$ by $3$ .

$108 - 36 - 22$

Step 14.2

Simplify by subtracting numbers.

Tap for more steps...

Step 14.2.1

Subtract $36$ from $108$ .

$72 - 22$

Step 14.2.2

Subtract $22$ from $72$ .

$50$

Step 15

$x = 3$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 3$ is a local minimum

Step 16

Find the y-value when $x = 3$ .

Tap for more steps...

Step 16.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{4} - 2 {(3)}^{3} - 11 {(3)}^{2} + 12 (3) + 36$

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

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

$f (3) = 81 - 2 {(3)}^{3} - 11 {(3)}^{2} + 12 (3) + 36$

Step 16.2.1.2

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

$f (3) = 81 - 2 \cdot 27 - 11 {(3)}^{2} + 12 (3) + 36$

Step 16.2.1.3

Multiply $- 2$ by $27$ .

$f (3) = 81 - 54 - 11 {(3)}^{2} + 12 (3) + 36$

Step 16.2.1.4

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

$f (3) = 81 - 54 - 11 \cdot 9 + 12 (3) + 36$

Step 16.2.1.5

Multiply $- 11$ by $9$ .

$f (3) = 81 - 54 - 99 + 12 (3) + 36$

Step 16.2.1.6

Multiply $12$ by $3$ .

$f (3) = 81 - 54 - 99 + 36 + 36$

Step 16.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 16.2.2.1

Subtract $54$ from $81$ .

$f (3) = 27 - 99 + 36 + 36$

Step 16.2.2.2

Subtract $99$ from $27$ .

$f (3) = - 72 + 36 + 36$

Step 16.2.2.3

Add $- 72$ and $36$ .

$f (3) = - 36 + 36$

Step 16.2.2.4

Add $- 36$ and $36$ .

$f (3) = 0$

Step 16.2.3

The final answer is $0$ .

$y = 0$

Step 17

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- 2)}^{2} - 12 \cdot - 2 - 22$

Step 18

Evaluate the second derivative.

Tap for more steps...

Step 18.1

Simplify each term.

Tap for more steps...

Step 18.1.1

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

$12 \cdot 4 - 12 \cdot - 2 - 22$

Step 18.1.2

Multiply $12$ by $4$ .

$48 - 12 \cdot - 2 - 22$

Step 18.1.3

Multiply $- 12$ by $- 2$ .

$48 + 24 - 22$

Step 18.2

Simplify by adding and subtracting.

Tap for more steps...

Step 18.2.1

Add $48$ and $24$ .

$72 - 22$

Step 18.2.2

Subtract $22$ from $72$ .

$50$

Step 19

$x = - 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 2$ is a local minimum

Step 20

Find the y-value when $x = - 2$ .

Tap for more steps...

Step 20.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = {(- 2)}^{4} - 2 {(- 2)}^{3} - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2

Simplify the result.

Tap for more steps...

Step 20.2.1

Simplify each term.

Tap for more steps...

Step 20.2.1.1

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

$f (- 2) = 16 - 2 {(- 2)}^{3} - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2.1.2

Multiply $- 2$ by ${(- 2)}^{3}$ by adding the exponents.

Tap for more steps...

Step 20.2.1.2.1

Multiply $- 2$ by ${(- 2)}^{3}$ .

Tap for more steps...

Step 20.2.1.2.1.1

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

$f (- 2) = 16 + (- 2) {(- 2)}^{3} - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2.1.2.1.2

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

$f (- 2) = 16 + {(- 2)}^{1 + 3} - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2.1.2.2

Add $1$ and $3$ .

$f (- 2) = 16 + {(- 2)}^{4} - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2.1.3

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

$f (- 2) = 16 + 16 - 11 {(- 2)}^{2} + 12 (- 2) + 36$

Step 20.2.1.4

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

$f (- 2) = 16 + 16 - 11 \cdot 4 + 12 (- 2) + 36$

Step 20.2.1.5

Multiply $- 11$ by $4$ .

$f (- 2) = 16 + 16 - 44 + 12 (- 2) + 36$

Step 20.2.1.6

Multiply $12$ by $- 2$ .

$f (- 2) = 16 + 16 - 44 - 24 + 36$

Step 20.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 20.2.2.1

Add $16$ and $16$ .

$f (- 2) = 32 - 44 - 24 + 36$

Step 20.2.2.2

Subtract $44$ from $32$ .

$f (- 2) = - 12 - 24 + 36$

Step 20.2.2.3

Subtract $24$ from $- 12$ .

$f (- 2) = - 36 + 36$

Step 20.2.2.4

Add $- 36$ and $36$ .

$f (- 2) = 0$

Step 20.2.3

The final answer is $0$ .

$y = 0$

Step 21

These are the local extrema for $f (x) = x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36$ .

$(\frac{1}{2}, \frac{625}{16})$ is a local maxima

$(3, 0)$ is a local minima

$(- 2, 0)$ is a local minima

Step 22