Calculus Examples

$x^{4} - 12 x^{3} + 48 x^{2} - 64 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 12 x^{3} + 48 x^{2} - 64 x$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.1.2

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

$4 x^{3} + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.1.2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - 12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.2.2

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

$4 x^{3} - 12 (3 x^{2}) + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.2.3

Multiply $3$ by $- 12$ .

$4 x^{3} - 36 x^{2} + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.3

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

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Step 1.1.3.1

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

$4 x^{3} - 36 x^{2} + 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 64 x]$

Step 1.1.3.2

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

$4 x^{3} - 36 x^{2} + 48 (2 x) + \frac{d}{d x} [- 64 x]$

Step 1.1.3.3

Multiply $2$ by $48$ .

$4 x^{3} - 36 x^{2} + 96 x + \frac{d}{d x} [- 64 x]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 64 x]$ .

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Step 1.1.4.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 x$ with respect to $x$ is $- 64 \frac{d}{d x} [x]$ .

$4 x^{3} - 36 x^{2} + 96 x - 64 \frac{d}{d x} [x]$

Step 1.1.4.2

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

$4 x^{3} - 36 x^{2} + 96 x - 64 \cdot 1$

Step 1.1.4.3

Multiply $- 64$ by $1$ .

$f' (x) = 4 x^{3} - 36 x^{2} + 96 x - 64$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 36 x^{2} + 96 x - 64$ .

$4 x^{3} - 36 x^{2} + 96 x - 64$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 36 x^{2} + 96 x - 64 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 36 x^{2} + 96 x - 64 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $4$ out of $4 x^{3} - 36 x^{2} + 96 x - 64$ .

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Step 2.2.1.1

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

$4 (x^{3}) - 36 x^{2} + 96 x - 64 = 0$

Step 2.2.1.2

Factor $4$ out of $- 36 x^{2}$ .

$4 (x^{3}) + 4 (- 9 x^{2}) + 96 x - 64 = 0$

Step 2.2.1.3

Factor $4$ out of $96 x$ .

$4 (x^{3}) + 4 (- 9 x^{2}) + 4 (24 x) - 64 = 0$

Step 2.2.1.4

Factor $4$ out of $- 64$ .

$4 x^{3} + 4 (- 9 x^{2}) + 4 (24 x) + 4 \cdot - 16 = 0$

Step 2.2.1.5

Factor $4$ out of $4 x^{3} + 4 (- 9 x^{2})$ .

$4 (x^{3} - 9 x^{2}) + 4 (24 x) + 4 \cdot - 16 = 0$

Step 2.2.1.6

Factor $4$ out of $4 (x^{3} - 9 x^{2}) + 4 (24 x)$ .

$4 (x^{3} - 9 x^{2} + 24 x) + 4 \cdot - 16 = 0$

Step 2.2.1.7

Factor $4$ out of $4 (x^{3} - 9 x^{2} + 24 x) + 4 \cdot - 16$ .

$4 (x^{3} - 9 x^{2} + 24 x - 16) = 0$

Step 2.2.2

Factor $x^{3} - 9 x^{2} + 24 x - 16$ 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.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

$q = \pm 1$

Step 2.2.2.2

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

$\pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Step 2.2.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.2.2.3.1

Substitute $1$ into the polynomial.

$1^{3} - 9 \cdot 1^{2} + 24 \cdot 1 - 16$

Step 2.2.2.3.2

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

$1 - 9 \cdot 1^{2} + 24 \cdot 1 - 16$

Step 2.2.2.3.3

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

$1 - 9 \cdot 1 + 24 \cdot 1 - 16$

Step 2.2.2.3.4

Multiply $- 9$ by $1$ .

$1 - 9 + 24 \cdot 1 - 16$

Step 2.2.2.3.5

Subtract $9$ from $1$ .

$- 8 + 24 \cdot 1 - 16$

Step 2.2.2.3.6

Multiply $24$ by $1$ .

$- 8 + 24 - 16$

Step 2.2.2.3.7

Add $- 8$ and $24$ .

$16 - 16$

Step 2.2.2.3.8

Subtract $16$ from $16$ .

$0$

Step 2.2.2.4

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

$\frac{x^{3} - 9 x^{2} + 24 x - 16}{x - 1}$

Step 2.2.2.5

Divide $x^{3} - 9 x^{2} + 24 x - 16$ by $x - 1$ .

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Step 2.2.2.5.1

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

$x$

$1$

$x^{3}$

$9 x^{2}$

$24 x$

$16$

Step 2.2.2.5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$

Step 2.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			+	$x^{3}$	-	$x^{2}$

Step 2.2.2.5.4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$

Step 2.2.2.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$

Step 2.2.2.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$

Step 2.2.2.5.7

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$

Step 2.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	+	$8 x$

Step 2.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 8 x$

				$x^{2}$	-	$8 x$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$

Step 2.2.2.5.10

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

				$x^{2}$	-	$8 x$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$

Step 2.2.2.5.11

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

				$x^{2}$	-	$8 x$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$	-	$16$

Step 2.2.2.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$	-	$16$

Step 2.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$	-	$16$
							+	$16 x$	-	$16$

Step 2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 16$

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$	-	$16$
							-	$16 x$	+	$16$

Step 2.2.2.5.15

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$1$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$16$
			-	$x^{3}$	+	$x^{2}$
					-	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	-	$8 x$
							+	$16 x$	-	$16$
							-	$16 x$	+	$16$
										$0$

Step 2.2.2.5.16

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

$x^{2} - 8 x + 16$

Step 2.2.2.6

Write $x^{3} - 9 x^{2} + 24 x - 16$ as a set of factors.

$4 ((x - 1) (x^{2} - 8 x + 16)) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor using the perfect square rule.

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Step 2.2.3.1.1

Rewrite $16$ as $4^{2}$ .

$4 ((x - 1) (x^{2} - 8 x + 4^{2})) = 0$

Step 2.2.3.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 2.2.3.1.3

Rewrite the polynomial.

$4 ((x - 1) (x^{2} - 2 \cdot x \cdot 4 + 4^{2})) = 0$

Step 2.2.3.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 2.2.3.2

Remove unnecessary parentheses.

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

Step 2.3

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

$x - 1 = 0$

${(x - 4)}^{2} = 0$

Step 2.4

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

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Step 2.4.1

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

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set ${(x - 4)}^{2}$ equal to $0$ .

${(x - 4)}^{2} = 0$

Step 2.5.2

Solve ${(x - 4)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

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

$x - 4 = 0$

Step 2.5.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.6

The final solution is all the values that make $4 (x - 1) {(x - 4)}^{2} = 0$ true.

$x = 1, 4$

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 $x^{4} - 12 x^{3} + 48 x^{2} - 64 x$ 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 $x$ .

${(1)}^{4} - 12 {(1)}^{3} + 48 {(1)}^{2} - 64 \cdot 1$

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

One to any power is one.

$1 - 12 {(1)}^{3} + 48 {(1)}^{2} - 64 \cdot 1$

Step 4.1.2.1.2

One to any power is one.

$1 - 12 \cdot 1 + 48 {(1)}^{2} - 64 \cdot 1$

Step 4.1.2.1.3

Multiply $- 12$ by $1$ .

$1 - 12 + 48 {(1)}^{2} - 64 \cdot 1$

Step 4.1.2.1.4

One to any power is one.

$1 - 12 + 48 \cdot 1 - 64 \cdot 1$

Step 4.1.2.1.5

Multiply $48$ by $1$ .

$1 - 12 + 48 - 64 \cdot 1$

Step 4.1.2.1.6

Multiply $- 64$ by $1$ .

$1 - 12 + 48 - 64$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Subtract $12$ from $1$ .

$- 11 + 48 - 64$

Step 4.1.2.2.2

Add $- 11$ and $48$ .

$37 - 64$

Step 4.1.2.2.3

Subtract $64$ from $37$ .

$- 27$

Step 4.2

Evaluate at $x = 4$ .

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Step 4.2.1

Substitute $4$ for $x$ .

${(4)}^{4} - 12 {(4)}^{3} + 48 {(4)}^{2} - 64 \cdot 4$

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

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

$256 - 12 {(4)}^{3} + 48 {(4)}^{2} - 64 \cdot 4$

Step 4.2.2.1.2

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

$256 - 12 \cdot 64 + 48 {(4)}^{2} - 64 \cdot 4$

Step 4.2.2.1.3

Multiply $- 12$ by $64$ .

$256 - 768 + 48 {(4)}^{2} - 64 \cdot 4$

Step 4.2.2.1.4

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

$256 - 768 + 48 \cdot 16 - 64 \cdot 4$

Step 4.2.2.1.5

Multiply $48$ by $16$ .

$256 - 768 + 768 - 64 \cdot 4$

Step 4.2.2.1.6

Multiply $- 64$ by $4$ .

$256 - 768 + 768 - 256$

Step 4.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.1

Subtract $768$ from $256$ .

$- 512 + 768 - 256$

Step 4.2.2.2.2

Add $- 512$ and $768$ .

$256 - 256$

Step 4.2.2.2.3

Subtract $256$ from $256$ .

$0$

Step 4.3

List all of the points.

$(1, - 27), (4, 0)$

Step 5