Calculus Examples

$f (x) = 6 x^{4} - 16 x^{3} - 111 x^{2} + 120 x + 27$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $6 x^{4} - 16 x^{3} - 111 x^{2} + 120 x + 27$ with respect to $x$ is $\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$ .

$\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

Tap for more steps...

Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$6 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.2.2

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

$6 (4 x^{3}) + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.2.3

Multiply $4$ by $6$ .

$24 x^{3} + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.3

Evaluate $\frac{d}{d x} [- 16 x^{3}]$ .

Tap for more steps...

Step 1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x^{3}$ with respect to $x$ is $- 16 \frac{d}{d x} [x^{3}]$ .

$24 x^{3} - 16 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.3.2

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

$24 x^{3} - 16 (3 x^{2}) + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.3.3

Multiply $3$ by $- 16$ .

$24 x^{3} - 48 x^{2} + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.4

Evaluate $\frac{d}{d x} [- 111 x^{2}]$ .

Tap for more steps...

Step 1.4.1

Since $- 111$ is constant with respect to $x$ , the derivative of $- 111 x^{2}$ with respect to $x$ is $- 111 \frac{d}{d x} [x^{2}]$ .

$24 x^{3} - 48 x^{2} - 111 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.4.2

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

$24 x^{3} - 48 x^{2} - 111 (2 x) + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.4.3

Multiply $2$ by $- 111$ .

$24 x^{3} - 48 x^{2} - 222 x + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 1.5

Evaluate $\frac{d}{d x} [120 x]$ .

Tap for more steps...

Step 1.5.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x$ with respect to $x$ is $120 \frac{d}{d x} [x]$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 \frac{d}{d x} [x] + \frac{d}{d x} [27]$

Step 1.5.2

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

$24 x^{3} - 48 x^{2} - 222 x + 120 \cdot 1 + \frac{d}{d x} [27]$

Step 1.5.3

Multiply $120$ by $1$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 + \frac{d}{d x} [27]$

Step 1.6

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.6.1

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 + 0$

Step 1.6.2

Add $24 x^{3} - 48 x^{2} - 222 x + 120$ and $0$ .

$24 x^{3} - 48 x^{2} - 222 x + 120$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $24 x^{3} - 48 x^{2} - 222 x + 120$ with respect to $x$ is $\frac{d}{d x} [24 x^{3}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 222 x] + \frac{d}{d x} [120]$ .

$f'' (x) = \frac{d}{d x} (24 x^{3}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.2

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

Tap for more steps...

Step 2.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 24 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.2.2

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

$f'' (x) = 24 (3 x^{2}) + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.2.3

Multiply $3$ by $24$ .

$f'' (x) = 72 x^{2} + \frac{d}{d x} (- 48 x^{2}) + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.3

Evaluate $\frac{d}{d x} [- 48 x^{2}]$ .

Tap for more steps...

Step 2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x^{2}$ with respect to $x$ is $- 48 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 72 x^{2} - 48 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.3.2

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

$f'' (x) = 72 x^{2} - 48 (2 x) + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.3.3

Multiply $2$ by $- 48$ .

$f'' (x) = 72 x^{2} - 96 x + \frac{d}{d x} (- 222 x) + \frac{d}{d x} (120)$

Step 2.4

Evaluate $\frac{d}{d x} [- 222 x]$ .

Tap for more steps...

Step 2.4.1

Since $- 222$ is constant with respect to $x$ , the derivative of $- 222 x$ with respect to $x$ is $- 222 \frac{d}{d x} [x]$ .

$f'' (x) = 72 x^{2} - 96 x - 222 \frac{d}{d x} x + \frac{d}{d x} (120)$

Step 2.4.2

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

$f'' (x) = 72 x^{2} - 96 x - 222 \cdot 1 + \frac{d}{d x} (120)$

Step 2.4.3

Multiply $- 222$ by $1$ .

$f'' (x) = 72 x^{2} - 96 x - 222 + \frac{d}{d x} (120)$

Step 2.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.5.1

Since $120$ is constant with respect to $x$ , the derivative of $120$ with respect to $x$ is $0$ .

$f'' (x) = 72 x^{2} - 96 x - 222 + 0$

Step 2.5.2

Add $72 x^{2} - 96 x - 222$ and $0$ .

$f'' (x) = 72 x^{2} - 96 x - 222$

Step 3

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

$24 x^{3} - 48 x^{2} - 222 x + 120 = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

$\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

Tap for more steps...

Step 4.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$6 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.2.2

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

$6 (4 x^{3}) + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.2.3

Multiply $4$ by $6$ .

$24 x^{3} + \frac{d}{d x} [- 16 x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 16 x^{3}]$ .

Tap for more steps...

Step 4.1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x^{3}$ with respect to $x$ is $- 16 \frac{d}{d x} [x^{3}]$ .

$24 x^{3} - 16 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.3.2

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

$24 x^{3} - 16 (3 x^{2}) + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.3.3

Multiply $3$ by $- 16$ .

$24 x^{3} - 48 x^{2} + \frac{d}{d x} [- 111 x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.4

Evaluate $\frac{d}{d x} [- 111 x^{2}]$ .

Tap for more steps...

Step 4.1.4.1

Since $- 111$ is constant with respect to $x$ , the derivative of $- 111 x^{2}$ with respect to $x$ is $- 111 \frac{d}{d x} [x^{2}]$ .

$24 x^{3} - 48 x^{2} - 111 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.4.2

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

$24 x^{3} - 48 x^{2} - 111 (2 x) + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.4.3

Multiply $2$ by $- 111$ .

$24 x^{3} - 48 x^{2} - 222 x + \frac{d}{d x} [120 x] + \frac{d}{d x} [27]$

Step 4.1.5

Evaluate $\frac{d}{d x} [120 x]$ .

Tap for more steps...

Step 4.1.5.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x$ with respect to $x$ is $120 \frac{d}{d x} [x]$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 \frac{d}{d x} [x] + \frac{d}{d x} [27]$

Step 4.1.5.2

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

$24 x^{3} - 48 x^{2} - 222 x + 120 \cdot 1 + \frac{d}{d x} [27]$

Step 4.1.5.3

Multiply $120$ by $1$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 + \frac{d}{d x} [27]$

Step 4.1.6

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.1.6.1

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 + 0$

Step 4.1.6.2

Add $24 x^{3} - 48 x^{2} - 222 x + 120$ and $0$ .

$f' (x) = 24 x^{3} - 48 x^{2} - 222 x + 120$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{3} - 48 x^{2} - 222 x + 120$ .

$24 x^{3} - 48 x^{2} - 222 x + 120$

Step 5

Set the first derivative equal to $0$ then solve the equation $24 x^{3} - 48 x^{2} - 222 x + 120 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$24 x^{3} - 48 x^{2} - 222 x + 120 = 0$

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

Factor $6$ out of $24 x^{3} - 48 x^{2} - 222 x + 120$ .

Tap for more steps...

Step 5.2.1.1

Factor $6$ out of $24 x^{3}$ .

$6 (4 x^{3}) - 48 x^{2} - 222 x + 120 = 0$

Step 5.2.1.2

Factor $6$ out of $- 48 x^{2}$ .

$6 (4 x^{3}) + 6 (- 8 x^{2}) - 222 x + 120 = 0$

Step 5.2.1.3

Factor $6$ out of $- 222 x$ .

$6 (4 x^{3}) + 6 (- 8 x^{2}) + 6 (- 37 x) + 120 = 0$

Step 5.2.1.4

Factor $6$ out of $120$ .

$6 (4 x^{3}) + 6 (- 8 x^{2}) + 6 (- 37 x) + 6 (20) = 0$

Step 5.2.1.5

Factor $6$ out of $6 (4 x^{3}) + 6 (- 8 x^{2})$ .

$6 (4 x^{3} - 8 x^{2}) + 6 (- 37 x) + 6 (20) = 0$

Step 5.2.1.6

Factor $6$ out of $6 (4 x^{3} - 8 x^{2}) + 6 (- 37 x)$ .

$6 (4 x^{3} - 8 x^{2} - 37 x) + 6 (20) = 0$

Step 5.2.1.7

Factor $6$ out of $6 (4 x^{3} - 8 x^{2} - 37 x) + 6 (20)$ .

$6 (4 x^{3} - 8 x^{2} - 37 x + 20) = 0$

Step 5.2.2

Factor $4 x^{3} - 8 x^{2} - 37 x + 20$ using the rational roots test.

Tap for more steps...

Step 5.2.2.1

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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1, \pm 4, \pm 2$

Step 5.2.2.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 20, \pm 5, \pm 10, \pm 2, \pm 2.5, \pm 4, \pm 1.25$

Step 5.2.2.3

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 5.2.2.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} - 8 \cdot {0.5}^{2} - 37 \cdot 0.5 + 20$

Step 5.2.2.3.2

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

$4 \cdot 0.125 - 8 \cdot {0.5}^{2} - 37 \cdot 0.5 + 20$

Step 5.2.2.3.3

Multiply $4$ by $0.125$ .

$0.5 - 8 \cdot {0.5}^{2} - 37 \cdot 0.5 + 20$

Step 5.2.2.3.4

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

$0.5 - 8 \cdot 0.25 - 37 \cdot 0.5 + 20$

Step 5.2.2.3.5

Multiply $- 8$ by $0.25$ .

$0.5 - 2 - 37 \cdot 0.5 + 20$

Step 5.2.2.3.6

Subtract $2$ from $0.5$ .

$- 1.5 - 37 \cdot 0.5 + 20$

Step 5.2.2.3.7

Multiply $- 37$ by $0.5$ .

$- 1.5 - 18.5 + 20$

Step 5.2.2.3.8

Subtract $18.5$ from $- 1.5$ .

$- 20 + 20$

Step 5.2.2.3.9

Add $- 20$ and $20$ .

$0$

Step 5.2.2.4

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.

$\frac{4 x^{3} - 8 x^{2} - 37 x + 20}{2 x - 1}$

Step 5.2.2.5

Divide $4 x^{3} - 8 x^{2} - 37 x + 20$ by $2 x - 1$ .

Tap for more steps...

Step 5.2.2.5.1

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

$2 x$

$1$

$4 x^{3}$

$8 x^{2}$

$37 x$

$20$

Step 5.2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$

Step 5.2.2.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			+	$4 x^{3}$	-	$2 x^{2}$

Step 5.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 2 x^{2}$

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$

Step 5.2.2.5.5

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$

Step 5.2.2.5.6

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$

Step 5.2.2.5.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$3 x$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$

Step 5.2.2.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$3 x$

Step 5.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 3 x$

				$2 x^{2}$	-	$3 x$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$

Step 5.2.2.5.10

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

				$2 x^{2}$	-	$3 x$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$

Step 5.2.2.5.11

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

				$2 x^{2}$	-	$3 x$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$	+	$20$

Step 5.2.2.5.12

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$3 x$	-	$20$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$	+	$20$

Step 5.2.2.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	-	$20$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$	+	$20$
							-	$40 x$	+	$20$

Step 5.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x + 20$

				$2 x^{2}$	-	$3 x$	-	$20$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$	+	$20$
							+	$40 x$	-	$20$

Step 5.2.2.5.15

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

				$2 x^{2}$	-	$3 x$	-	$20$
$2 x$	-	$1$		$4 x^{3}$	-	$8 x^{2}$	-	$37 x$	+	$20$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$3 x$
							-	$40 x$	+	$20$
							+	$40 x$	-	$20$
										$0$

Step 5.2.2.5.16

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

$2 x^{2} - 3 x - 20$

Step 5.2.2.6

Write $4 x^{3} - 8 x^{2} - 37 x + 20$ as a set of factors.

$6 ((2 x - 1) (2 x^{2} - 3 x - 20)) = 0$

Step 5.2.3

Factor.

Tap for more steps...

Step 5.2.3.1

Factor by grouping.

Tap for more steps...

Step 5.2.3.1.1

Factor by grouping.

Tap for more steps...

Step 5.2.3.1.1.1

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 $a \cdot c = 2 \cdot - 20 = - 40$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 5.2.3.1.1.1.1

Factor $- 3$ out of $- 3 x$ .

$6 ((2 x - 1) (2 x^{2} - 3 x - 20)) = 0$

Step 5.2.3.1.1.1.2

Rewrite $- 3$ as $5$ plus $- 8$

$6 ((2 x - 1) (2 x^{2} + (5 - 8) x - 20)) = 0$

Step 5.2.3.1.1.1.3

Apply the distributive property.

$6 ((2 x - 1) (2 x^{2} + 5 x - 8 x - 20)) = 0$

Step 5.2.3.1.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.2.3.1.1.2.1

Group the first two terms and the last two terms.

$6 ((2 x - 1) ((2 x^{2} + 5 x) - 8 x - 20)) = 0$

Step 5.2.3.1.1.2.2

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

$6 ((2 x - 1) (x (2 x + 5) - 4 (2 x + 5))) = 0$

Step 5.2.3.1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$6 ((2 x - 1) ((2 x + 5) (x - 4))) = 0$

Step 5.2.3.1.2

Remove unnecessary parentheses.

$6 ((2 x - 1) (2 x + 5) (x - 4)) = 0$

Step 5.2.3.2

Remove unnecessary parentheses.

$6 (2 x - 1) (2 x + 5) (x - 4) = 0$

Step 5.3

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

$2 x - 1 = 0$

$2 x + 5 = 0$

$x - 4 = 0$

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

$2 x - 1 = 0$

Step 5.4.2

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

Tap for more steps...

Step 5.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 5.4.2.2

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

Tap for more steps...

Step 5.4.2.2.1

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

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

Step 5.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.2.2.2.1.1

Cancel the common factor.

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

Step 5.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.5

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Step 5.5.2

Solve $2 x + 5 = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Step 5.5.2.2

Divide each term in $2 x = - 5$ by $2$ and simplify.

Tap for more steps...

Step 5.5.2.2.1

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 5.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 5.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Step 5.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 5.6

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 5.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.7

The final solution is all the values that make $6 (2 x - 1) (2 x + 5) (x - 4) = 0$ true.

$x = \frac{1}{2}, - \frac{5}{2}, 4$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

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 7

Critical points to evaluate.

$x = \frac{1}{2}, - \frac{5}{2}, 4$

Step 8

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.

$72 {(\frac{1}{2})}^{2} - 96 (\frac{1}{2}) - 222$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

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

$72 \frac{1^{2}}{2^{2}} - 96 (\frac{1}{2}) - 222$

Step 9.1.2

One to any power is one.

$72 \frac{1}{2^{2}} - 96 (\frac{1}{2}) - 222$

Step 9.1.3

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

$72 (\frac{1}{4}) - 96 (\frac{1}{2}) - 222$

Step 9.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 9.1.4.1

Factor $4$ out of $72$ .

$4 (18) \frac{1}{4} - 96 (\frac{1}{2}) - 222$

Step 9.1.4.2

Cancel the common factor.

$4 \cdot 18 \frac{1}{4} - 96 (\frac{1}{2}) - 222$

Step 9.1.4.3

Rewrite the expression.

$18 - 96 (\frac{1}{2}) - 222$

Step 9.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1.5.1

Factor $2$ out of $- 96$ .

$18 + 2 (- 48) \frac{1}{2} - 222$

Step 9.1.5.2

Cancel the common factor.

$18 + 2 \cdot - 48 \frac{1}{2} - 222$

Step 9.1.5.3

Rewrite the expression.

$18 - 48 - 222$

Step 9.2

Simplify by subtracting numbers.

Tap for more steps...

Step 9.2.1

Subtract $48$ from $18$ .

$- 30 - 222$

Step 9.2.2

Subtract $222$ from $- 30$ .

$- 252$

Step 10

$x = \frac{1}{2}$ 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 11

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

Tap for more steps...

Step 11.1

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

$f (\frac{1}{2}) = 6 {(\frac{1}{2})}^{4} - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

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

$f (\frac{1}{2}) = 6 (\frac{1^{4}}{2^{4}}) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.2

One to any power is one.

$f (\frac{1}{2}) = 6 (\frac{1}{2^{4}}) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.3

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

$f (\frac{1}{2}) = 6 (\frac{1}{16}) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1.4.1

Factor $2$ out of $6$ .

$f (\frac{1}{2}) = 2 (3) (\frac{1}{16}) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.4.2

Factor $2$ out of $16$ .

$f (\frac{1}{2}) = 2 \cdot (3 (\frac{1}{2 \cdot 8})) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.4.3

Cancel the common factor.

$f (\frac{1}{2}) = 2 \cdot (3 (\frac{1}{2 \cdot 8})) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.4.4

Rewrite the expression.

$f (\frac{1}{2}) = 3 (\frac{1}{8}) - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.5

Combine $3$ and $\frac{1}{8}$ .

$f (\frac{1}{2}) = \frac{3}{8} - 16 {(\frac{1}{2})}^{3} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.6

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

$f (\frac{1}{2}) = \frac{3}{8} - 16 \frac{1^{3}}{2^{3}} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.7

One to any power is one.

$f (\frac{1}{2}) = \frac{3}{8} - 16 \frac{1}{2^{3}} - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.8

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

$f (\frac{1}{2}) = \frac{3}{8} - 16 (\frac{1}{8}) - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.9

Cancel the common factor of $8$ .

Tap for more steps...

Step 11.2.1.9.1

Factor $8$ out of $- 16$ .

$f (\frac{1}{2}) = \frac{3}{8} + 8 (- 2) (\frac{1}{8}) - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.9.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{3}{8} + 8 \cdot (- 2 (\frac{1}{8})) - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.9.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{3}{8} - 2 - 111 {(\frac{1}{2})}^{2} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.10

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

$f (\frac{1}{2}) = \frac{3}{8} - 2 - 111 \frac{1^{2}}{2^{2}} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.11

One to any power is one.

$f (\frac{1}{2}) = \frac{3}{8} - 2 - 111 \frac{1}{2^{2}} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.12

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

$f (\frac{1}{2}) = \frac{3}{8} - 2 - 111 (\frac{1}{4}) + 120 (\frac{1}{2}) + 27$

Step 11.2.1.13

Combine $- 111$ and $\frac{1}{4}$ .

$f (\frac{1}{2}) = \frac{3}{8} - 2 + \frac{- 111}{4} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.14

Move the negative in front of the fraction.

$f (\frac{1}{2}) = \frac{3}{8} - 2 - \frac{111}{4} + 120 (\frac{1}{2}) + 27$

Step 11.2.1.15

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1.15.1

Factor $2$ out of $120$ .

$f (\frac{1}{2}) = \frac{3}{8} - 2 - \frac{111}{4} + 2 (60) (\frac{1}{2}) + 27$

Step 11.2.1.15.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{3}{8} - 2 - \frac{111}{4} + 2 \cdot (60 (\frac{1}{2})) + 27$

Step 11.2.1.15.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{3}{8} - 2 - \frac{111}{4} + 60 + 27$

Step 11.2.2

Find the common denominator.

Tap for more steps...

Step 11.2.2.1

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

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2}{1} - \frac{111}{4} + 60 + 27$

Step 11.2.2.2

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

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2}{1} \cdot \frac{8}{8} - \frac{111}{4} + 60 + 27$

Step 11.2.2.3

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

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111}{4} + 60 + 27$

Step 11.2.2.4

Multiply $\frac{111}{4}$ by $\frac{2}{2}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - (\frac{111}{4} \cdot \frac{2}{2}) + 60 + 27$

Step 11.2.2.5

Multiply $\frac{111}{4}$ by $\frac{2}{2}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + 60 + 27$

Step 11.2.2.6

Write $60$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60}{1} + 27$

Step 11.2.2.7

Multiply $\frac{60}{1}$ by $\frac{8}{8}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60}{1} \cdot \frac{8}{8} + 27$

Step 11.2.2.8

Multiply $\frac{60}{1}$ by $\frac{8}{8}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60 \cdot 8}{8} + 27$

Step 11.2.2.9

Write $27$ as a fraction with denominator $1$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60 \cdot 8}{8} + \frac{27}{1}$

Step 11.2.2.10

Multiply $\frac{27}{1}$ by $\frac{8}{8}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60 \cdot 8}{8} + \frac{27}{1} \cdot \frac{8}{8}$

Step 11.2.2.11

Multiply $\frac{27}{1}$ by $\frac{8}{8}$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{4 \cdot 2} + \frac{60 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 11.2.2.12

Reorder the factors of $4 \cdot 2$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{2 \cdot 4} + \frac{60 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 11.2.2.13

Multiply $2$ by $4$ .

$f (\frac{1}{2}) = \frac{3}{8} + \frac{- 2 \cdot 8}{8} - \frac{111 \cdot 2}{8} + \frac{60 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 11.2.3

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{3 - 2 \cdot 8 - 111 \cdot 2 + 60 \cdot 8 + 27 \cdot 8}{8}$

Step 11.2.4

Simplify each term.

Tap for more steps...

Step 11.2.4.1

Multiply $- 2$ by $8$ .

$f (\frac{1}{2}) = \frac{3 - 16 - 111 \cdot 2 + 60 \cdot 8 + 27 \cdot 8}{8}$

Step 11.2.4.2

Multiply $- 111$ by $2$ .

$f (\frac{1}{2}) = \frac{3 - 16 - 222 + 60 \cdot 8 + 27 \cdot 8}{8}$

Step 11.2.4.3

Multiply $60$ by $8$ .

$f (\frac{1}{2}) = \frac{3 - 16 - 222 + 480 + 27 \cdot 8}{8}$

Step 11.2.4.4

Multiply $27$ by $8$ .

$f (\frac{1}{2}) = \frac{3 - 16 - 222 + 480 + 216}{8}$

Step 11.2.5

Simplify by adding and subtracting.

Tap for more steps...

Step 11.2.5.1

Subtract $16$ from $3$ .

$f (\frac{1}{2}) = \frac{- 13 - 222 + 480 + 216}{8}$

Step 11.2.5.2

Subtract $222$ from $- 13$ .

$f (\frac{1}{2}) = \frac{- 235 + 480 + 216}{8}$

Step 11.2.5.3

Add $- 235$ and $480$ .

$f (\frac{1}{2}) = \frac{245 + 216}{8}$

Step 11.2.5.4

Add $245$ and $216$ .

$f (\frac{1}{2}) = \frac{461}{8}$

Step 11.2.6

The final answer is $\frac{461}{8}$ .

$y = \frac{461}{8}$

Step 12

Evaluate the second derivative at $x = - \frac{5}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$72 {(- \frac{5}{2})}^{2} - 96 (- \frac{5}{2}) - 222$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 13.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$72 ({(- 1)}^{2} {(\frac{5}{2})}^{2}) - 96 (- \frac{5}{2}) - 222$

Step 13.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$72 ({(- 1)}^{2} \frac{5^{2}}{2^{2}}) - 96 (- \frac{5}{2}) - 222$

Step 13.1.2

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

$72 (1 \frac{5^{2}}{2^{2}}) - 96 (- \frac{5}{2}) - 222$

Step 13.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$72 \frac{5^{2}}{2^{2}} - 96 (- \frac{5}{2}) - 222$

Step 13.1.4

Raise $5$ to the power of $2$ .

$72 \frac{25}{2^{2}} - 96 (- \frac{5}{2}) - 222$

Step 13.1.5

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

$72 (\frac{25}{4}) - 96 (- \frac{5}{2}) - 222$

Step 13.1.6

Cancel the common factor of $4$ .

Tap for more steps...

Step 13.1.6.1

Factor $4$ out of $72$ .

$4 (18) \frac{25}{4} - 96 (- \frac{5}{2}) - 222$

Step 13.1.6.2

Cancel the common factor.

$4 \cdot 18 \frac{25}{4} - 96 (- \frac{5}{2}) - 222$

Step 13.1.6.3

Rewrite the expression.

$18 \cdot 25 - 96 (- \frac{5}{2}) - 222$

Step 13.1.7

Multiply $18$ by $25$ .

$450 - 96 (- \frac{5}{2}) - 222$

Step 13.1.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.1.8.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$450 - 96 (\frac{- 5}{2}) - 222$

Step 13.1.8.2

Factor $2$ out of $- 96$ .

$450 + 2 (- 48) \frac{- 5}{2} - 222$

Step 13.1.8.3

Cancel the common factor.

$450 + 2 \cdot - 48 \frac{- 5}{2} - 222$

Step 13.1.8.4

Rewrite the expression.

$450 - 48 \cdot - 5 - 222$

Step 13.1.9

Multiply $- 48$ by $- 5$ .

$450 + 240 - 222$

Step 13.2

Simplify by adding and subtracting.

Tap for more steps...

Step 13.2.1

Add $450$ and $240$ .

$690 - 222$

Step 13.2.2

Subtract $222$ from $690$ .

$468$

Step 14

$x = - \frac{5}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{5}{2}$ is a local minimum

Step 15

Find the y-value when $x = - \frac{5}{2}$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f (- \frac{5}{2}) = 6 {(- \frac{5}{2})}^{4} - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = 6 ({(- 1)}^{4} {(\frac{5}{2})}^{4}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = 6 ({(- 1)}^{4} (\frac{5^{4}}{2^{4}})) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.2

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

$f (- \frac{5}{2}) = 6 (1 (\frac{5^{4}}{2^{4}})) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.3

Multiply $\frac{5^{4}}{2^{4}}$ by $1$ .

$f (- \frac{5}{2}) = 6 (\frac{5^{4}}{2^{4}}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.4

Raise $5$ to the power of $4$ .

$f (- \frac{5}{2}) = 6 (\frac{625}{2^{4}}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.5

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

$f (- \frac{5}{2}) = 6 (\frac{625}{16}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.2.1.6.1

Factor $2$ out of $6$ .

$f (- \frac{5}{2}) = 2 (3) (\frac{625}{16}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.6.2

Factor $2$ out of $16$ .

$f (- \frac{5}{2}) = 2 \cdot (3 (\frac{625}{2 \cdot 8})) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.6.3

Cancel the common factor.

$f (- \frac{5}{2}) = 2 \cdot (3 (\frac{625}{2 \cdot 8})) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.6.4

Rewrite the expression.

$f (- \frac{5}{2}) = 3 (\frac{625}{8}) - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.7

Combine $3$ and $\frac{625}{8}$ .

$f (- \frac{5}{2}) = \frac{3 \cdot 625}{8} - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.8

Multiply $3$ by $625$ .

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 {(- \frac{5}{2})}^{3} - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.1.9.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 ({(- 1)}^{3} {(\frac{5}{2})}^{3}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.9.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 ({(- 1)}^{3} (\frac{5^{3}}{2^{3}})) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.10

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

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 (- \frac{5^{3}}{2^{3}}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.11

Raise $5$ to the power of $3$ .

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 (- \frac{125}{2^{3}}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.12

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

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 (- \frac{125}{8}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.13

Cancel the common factor of $8$ .

Tap for more steps...

Step 15.2.1.13.1

Move the leading negative in $- \frac{125}{8}$ into the numerator.

$f (- \frac{5}{2}) = \frac{1875}{8} - 16 (\frac{- 125}{8}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.13.2

Factor $8$ out of $- 16$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 8 (- 2) (\frac{- 125}{8}) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.13.3

Cancel the common factor.

$f (- \frac{5}{2}) = \frac{1875}{8} + 8 \cdot (- 2 (\frac{- 125}{8})) - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.13.4

Rewrite the expression.

$f (- \frac{5}{2}) = \frac{1875}{8} - 2 \cdot - 125 - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.14

Multiply $- 2$ by $- 125$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 {(- \frac{5}{2})}^{2} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.15

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.1.15.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 ({(- 1)}^{2} {(\frac{5}{2})}^{2}) + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.15.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 ({(- 1)}^{2} (\frac{5^{2}}{2^{2}})) + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.16

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

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 (1 (\frac{5^{2}}{2^{2}})) + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.17

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 \frac{5^{2}}{2^{2}} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.18

Raise $5$ to the power of $2$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 \frac{25}{2^{2}} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.19

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

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - 111 (\frac{25}{4}) + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.20

Multiply $- 111 (\frac{25}{4})$ .

Tap for more steps...

Step 15.2.1.20.1

Combine $- 111$ and $\frac{25}{4}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 + \frac{- 111 \cdot 25}{4} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.20.2

Multiply $- 111$ by $25$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 + \frac{- 2775}{4} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.21

Move the negative in front of the fraction.

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} + 120 (- \frac{5}{2}) + 27$

Step 15.2.1.22

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.2.1.22.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} + 120 (\frac{- 5}{2}) + 27$

Step 15.2.1.22.2

Factor $2$ out of $120$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} + 2 (60) (\frac{- 5}{2}) + 27$

Step 15.2.1.22.3

Cancel the common factor.

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} + 2 \cdot (60 (\frac{- 5}{2})) + 27$

Step 15.2.1.22.4

Rewrite the expression.

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} + 60 \cdot - 5 + 27$

Step 15.2.1.23

Multiply $60$ by $- 5$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + 250 - \frac{2775}{4} - 300 + 27$

Step 15.2.2

Find the common denominator.

Tap for more steps...

Step 15.2.2.1

Write $250$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250}{1} - \frac{2775}{4} - 300 + 27$

Step 15.2.2.2

Multiply $\frac{250}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250}{1} \cdot \frac{8}{8} - \frac{2775}{4} - 300 + 27$

Step 15.2.2.3

Multiply $\frac{250}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775}{4} - 300 + 27$

Step 15.2.2.4

Multiply $\frac{2775}{4}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - (\frac{2775}{4} \cdot \frac{2}{2}) - 300 + 27$

Step 15.2.2.5

Multiply $\frac{2775}{4}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} - 300 + 27$

Step 15.2.2.6

Write $- 300$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300}{1} + 27$

Step 15.2.2.7

Multiply $\frac{- 300}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300}{1} \cdot \frac{8}{8} + 27$

Step 15.2.2.8

Multiply $\frac{- 300}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300 \cdot 8}{8} + 27$

Step 15.2.2.9

Write $27$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300 \cdot 8}{8} + \frac{27}{1}$

Step 15.2.2.10

Multiply $\frac{27}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300 \cdot 8}{8} + \frac{27}{1} \cdot \frac{8}{8}$

Step 15.2.2.11

Multiply $\frac{27}{1}$ by $\frac{8}{8}$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{4 \cdot 2} + \frac{- 300 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 15.2.2.12

Reorder the factors of $4 \cdot 2$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{2 \cdot 4} + \frac{- 300 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 15.2.2.13

Multiply $2$ by $4$ .

$f (- \frac{5}{2}) = \frac{1875}{8} + \frac{250 \cdot 8}{8} - \frac{2775 \cdot 2}{8} + \frac{- 300 \cdot 8}{8} + \frac{27 \cdot 8}{8}$

Step 15.2.3

Combine the numerators over the common denominator.

$f (- \frac{5}{2}) = \frac{1875 + 250 \cdot 8 - 2775 \cdot 2 - 300 \cdot 8 + 27 \cdot 8}{8}$

Step 15.2.4

Simplify each term.

Tap for more steps...

Step 15.2.4.1

Multiply $250$ by $8$ .

$f (- \frac{5}{2}) = \frac{1875 + 2000 - 2775 \cdot 2 - 300 \cdot 8 + 27 \cdot 8}{8}$

Step 15.2.4.2

Multiply $- 2775$ by $2$ .

$f (- \frac{5}{2}) = \frac{1875 + 2000 - 5550 - 300 \cdot 8 + 27 \cdot 8}{8}$

Step 15.2.4.3

Multiply $- 300$ by $8$ .

$f (- \frac{5}{2}) = \frac{1875 + 2000 - 5550 - 2400 + 27 \cdot 8}{8}$

Step 15.2.4.4

Multiply $27$ by $8$ .

$f (- \frac{5}{2}) = \frac{1875 + 2000 - 5550 - 2400 + 216}{8}$

Step 15.2.5

Simplify the expression.

Tap for more steps...

Step 15.2.5.1

Add $1875$ and $2000$ .

$f (- \frac{5}{2}) = \frac{3875 - 5550 - 2400 + 216}{8}$

Step 15.2.5.2

Subtract $5550$ from $3875$ .

$f (- \frac{5}{2}) = \frac{- 1675 - 2400 + 216}{8}$

Step 15.2.5.3

Subtract $2400$ from $- 1675$ .

$f (- \frac{5}{2}) = \frac{- 4075 + 216}{8}$

Step 15.2.5.4

Add $- 4075$ and $216$ .

$f (- \frac{5}{2}) = \frac{- 3859}{8}$

Step 15.2.5.5

Move the negative in front of the fraction.

$f (- \frac{5}{2}) = - \frac{3859}{8}$

Step 15.2.6

The final answer is $- \frac{3859}{8}$ .

$y = - \frac{3859}{8}$

Step 16

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$72 {(4)}^{2} - 96 \cdot 4 - 222$

Step 17

Evaluate the second derivative.

Tap for more steps...

Step 17.1

Simplify each term.

Tap for more steps...

Step 17.1.1

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

$72 \cdot 16 - 96 \cdot 4 - 222$

Step 17.1.2

Multiply $72$ by $16$ .

$1152 - 96 \cdot 4 - 222$

Step 17.1.3

Multiply $- 96$ by $4$ .

$1152 - 384 - 222$

Step 17.2

Simplify by subtracting numbers.

Tap for more steps...

Step 17.2.1

Subtract $384$ from $1152$ .

$768 - 222$

Step 17.2.2

Subtract $222$ from $768$ .

$546$

Step 18

$x = 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4$ is a local minimum

Step 19

Find the y-value when $x = 4$ .

Tap for more steps...

Step 19.1

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

$f (4) = 6 {(4)}^{4} - 16 {(4)}^{3} - 111 {(4)}^{2} + 120 (4) + 27$

Step 19.2

Simplify the result.

Tap for more steps...

Step 19.2.1

Simplify each term.

Tap for more steps...

Step 19.2.1.1

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

$f (4) = 6 \cdot 256 - 16 {(4)}^{3} - 111 {(4)}^{2} + 120 (4) + 27$

Step 19.2.1.2

Multiply $6$ by $256$ .

$f (4) = 1536 - 16 {(4)}^{3} - 111 {(4)}^{2} + 120 (4) + 27$

Step 19.2.1.3

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

$f (4) = 1536 - 16 \cdot 64 - 111 {(4)}^{2} + 120 (4) + 27$

Step 19.2.1.4

Multiply $- 16$ by $64$ .

$f (4) = 1536 - 1024 - 111 {(4)}^{2} + 120 (4) + 27$

Step 19.2.1.5

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

$f (4) = 1536 - 1024 - 111 \cdot 16 + 120 (4) + 27$

Step 19.2.1.6

Multiply $- 111$ by $16$ .

$f (4) = 1536 - 1024 - 1776 + 120 (4) + 27$

Step 19.2.1.7

Multiply $120$ by $4$ .

$f (4) = 1536 - 1024 - 1776 + 480 + 27$

Step 19.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 19.2.2.1

Subtract $1024$ from $1536$ .

$f (4) = 512 - 1776 + 480 + 27$

Step 19.2.2.2

Subtract $1776$ from $512$ .

$f (4) = - 1264 + 480 + 27$

Step 19.2.2.3

Add $- 1264$ and $480$ .

$f (4) = - 784 + 27$

Step 19.2.2.4

Add $- 784$ and $27$ .

$f (4) = - 757$

Step 19.2.3

The final answer is $- 757$ .

$y = - 757$

Step 20

These are the local extrema for $f (x) = 6 x^{4} - 16 x^{3} - 111 x^{2} + 120 x + 27$ .

$(\frac{1}{2}, \frac{461}{8})$ is a local maxima

$(- \frac{5}{2}, - \frac{3859}{8})$ is a local minima

$(4, - 757)$ is a local minima

Step 21