Calculus Examples

g (x) = \frac{6 x^{2}}{x - 2}

$g (x) = \frac{6 x^{2}}{x - 2}$ ,

[- 2, 1]

$[- 2, 1]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

\frac{6 x^{2}}{x - 2}

$\frac{6 x^{2}}{x - 2}$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.1.2

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = x - 2

$g (x) = x - 2$ .

6 \frac{(x - 2) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}

$6 \frac{(x - 2) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

6 \frac{(x - 2) (2 x) - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}

$6 \frac{(x - 2) (2 x) - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3.2

Move

2

$2$ to the left of

x - 2

$x - 2$ .

6 \frac{2 \cdot (x - 2) x - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}

$6 \frac{2 \cdot (x - 2) x - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3.3

By the Sum Rule, the derivative of

x - 2

$x - 2$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 2]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

6 \frac{2 (x - 2) x - x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}

$6 \frac{2 (x - 2) x - x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.1.1.3.4

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

6 \frac{2 (x - 2) x - x^{2} (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}

$6 \frac{2 (x - 2) x - x^{2} (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.1.1.3.5

Since

- 2

$- 2$ is constant with respect to

x

$x$ , the derivative of

- 2

$- 2$ with respect to

x

$x$ is

0

$0$ .

6 \frac{2 (x - 2) x - x^{2} (1 + 0)}{{(x - 2)}^{2}}

$6 \frac{2 (x - 2) x - x^{2} (1 + 0)}{{(x - 2)}^{2}}$

Step 1.1.1.3.6

Combine fractions.

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Step 1.1.1.3.6.1

Add

1

$1$ and

0

$0$ .

6 \frac{2 (x - 2) x - x^{2} \cdot 1}{{(x - 2)}^{2}}

$6 \frac{2 (x - 2) x - x^{2} \cdot 1}{{(x - 2)}^{2}}$

Step 1.1.1.3.6.2

Multiply

- 1

$- 1$ by

1

$1$ .

6 \frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}

$6 \frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}$

Step 1.1.1.3.6.3

Combine

6

$6$ and

\frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}

$\frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}$ .

\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}$

\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}$

\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Apply the distributive property.

\frac{6 ((2 x + 2 \cdot - 2) x - x^{2})}{{(x - 2)}^{2}}

$\frac{6 ((2 x + 2 \cdot - 2) x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.2

Apply the distributive property.

\frac{6 (2 x \cdot x + 2 \cdot - 2 x - x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 x \cdot x + 2 \cdot - 2 x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.3

Apply the distributive property.

\frac{6 (2 x \cdot x) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 x \cdot x) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4

Simplify the numerator.

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Step 1.1.1.4.4.1

Simplify each term.

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Step 1.1.1.4.4.1.1

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 1.1.1.4.4.1.1.1

Move

x

$x$ .

\frac{6 (2 (x \cdot x)) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 (x \cdot x)) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.1.2

Multiply

x

$x$ by

x

$x$ .

\frac{6 (2 x^{2}) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 x^{2}) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

\frac{6 (2 x^{2}) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{6 (2 x^{2}) + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.2

Multiply

2

$2$ by

6

$6$ .

\frac{12 x^{2} + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{12 x^{2} + 6 (2 \cdot - 2 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.3

Multiply

2

$2$ by

- 2

$- 2$ .

\frac{12 x^{2} + 6 (- 4 x) + 6 (- x^{2})}{{(x - 2)}^{2}}

$\frac{12 x^{2} + 6 (- 4 x) + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.4

Multiply

$- 4$ by

$6$ .

$\frac{12 x^{2} - 24 x + 6 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.5

Multiply

$- 1$ by

$6$ .

$\frac{12 x^{2} - 24 x - 6 x^{2}}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.2

Subtract

$6 x^{2}$ from

$12 x^{2}$ .

$\frac{6 x^{2} - 24 x}{{(x - 2)}^{2}}$

Step 1.1.1.4.5

Factor

$6 x$ out of

$6 x^{2} - 24 x$ .

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Step 1.1.1.4.5.1

Factor

$6 x$ out of

$6 x^{2}$ .

$\frac{6 x (x) - 24 x}{{(x - 2)}^{2}}$

Step 1.1.1.4.5.2

Factor

$6 x$ out of

$- 24 x$ .

$\frac{6 x (x) + 6 x (- 4)}{{(x - 2)}^{2}}$

Step 1.1.1.4.5.3

Factor

$6 x$ out of

$6 x (x) + 6 x (- 4)$ .

$f' (x) = \frac{6 x (x - 4)}{{(x - 2)}^{2}}$

Step 1.1.2

The first derivative of

$g (x)$ with respect to

$x$ is

$\frac{6 x (x - 4)}{{(x - 2)}^{2}}$ .

$\frac{6 x (x - 4)}{{(x - 2)}^{2}}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{6 x (x - 4)}{{(x - 2)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{6 x (x - 4)}{{(x - 2)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$6 x (x - 4) = 0$

Step 1.2.3

Solve the equation for

$x$ .

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Step 1.2.3.1

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$x - 4 = 0$

Step 1.2.3.2

Set

$x$ equal to

$0$ .

$x = 0$

Step 1.2.3.3

Set

$x - 4$ equal to

$0$ and solve for

$x$ .

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Step 1.2.3.3.1

Set

$x - 4$ equal to

$0$ .

$x - 4 = 0$

Step 1.2.3.3.2

Add

$4$ to both sides of the equation.

$x = 4$

Step 1.2.3.4

The final solution is all the values that make

$6 x (x - 4) = 0$ true.

$x = 0, 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in

$\frac{6 x (x - 4)}{{(x - 2)}^{2}}$ equal to

$0$ to find where the expression is undefined.

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

Step 1.3.2

Solve for

$x$ .

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Step 1.3.2.1

Set the

$x - 2$ equal to

$0$ .

$x - 2 = 0$

Step 1.3.2.2

Add

$2$ to both sides of the equation.

$x = 2$

Step 1.4

Evaluate

$\frac{6 x^{2}}{x - 2}$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 0$ .

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Step 1.4.1.1

Substitute

$0$ for

$x$ .

$\frac{6 {(0)}^{2}}{(0) - 2}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Cancel the common factor of

$6$ and

$(0) - 2$ .

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Step 1.4.1.2.1.1

Factor

$2$ out of

$6 {(0)}^{2}$ .

$\frac{2 (3 {(0)}^{2})}{(0) - 2}$

Step 1.4.1.2.1.2

Cancel the common factors.

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Step 1.4.1.2.1.2.1

Factor

$2$ out of

$0$ .

$\frac{2 (3 {(0)}^{2})}{2 (0) - 2}$

Step 1.4.1.2.1.2.2

Factor

$2$ out of

$- 2$ .

$\frac{2 (3 {(0)}^{2})}{2 \cdot 0 + 2 \cdot - 1}$

Step 1.4.1.2.1.2.3

Factor

$2$ out of

$2 \cdot 0 + 2 \cdot - 1$ .

$\frac{2 (3 {(0)}^{2})}{2 \cdot (0 - 1)}$

Step 1.4.1.2.1.2.4

Cancel the common factor.

$\frac{2 (3 {(0)}^{2})}{2 \cdot (0 - 1)}$

Step 1.4.1.2.1.2.5

Rewrite the expression.

$\frac{3 {(0)}^{2}}{0 - 1}$

Step 1.4.1.2.2

Simplify the expression.

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Step 1.4.1.2.2.1

Raising

$0$ to any positive power yields

$0$ .

$\frac{3 \cdot 0}{0 - 1}$

Step 1.4.1.2.2.2

Subtract

$1$ from

$0$ .

$\frac{3 \cdot 0}{- 1}$

Step 1.4.1.2.2.3

Multiply

$3$ by

$0$ .

$\frac{0}{- 1}$

Step 1.4.1.2.2.4

Divide

$0$ by

$- 1$ .

$0$

Step 1.4.2

Evaluate at

$x = 4$ .

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Step 1.4.2.1

Substitute

$4$ for

$x$ .

$\frac{6 {(4)}^{2}}{(4) - 2}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Cancel the common factor of

$6$ and

$(4) - 2$ .

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Step 1.4.2.2.1.1

Factor

$2$ out of

$6 {(4)}^{2}$ .

$\frac{2 (3 {(4)}^{2})}{(4) - 2}$

Step 1.4.2.2.1.2

Cancel the common factors.

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Step 1.4.2.2.1.2.1

Factor

$2$ out of

$4$ .

$\frac{2 (3 {(4)}^{2})}{2 (2) - 2}$

Step 1.4.2.2.1.2.2

Factor

$2$ out of

$- 2$ .

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

Step 1.4.2.2.1.2.3

Factor

$2$ out of

$2 \cdot 2 + 2 \cdot - 1$ .

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

Step 1.4.2.2.1.2.4

Cancel the common factor.

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

Step 1.4.2.2.1.2.5

Rewrite the expression.

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

Step 1.4.2.2.2

Simplify the expression.

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Step 1.4.2.2.2.1

Raise

$4$ to the power of

$2$ .

$\frac{3 \cdot 16}{2 - 1}$

Step 1.4.2.2.2.2

Subtract

$1$ from

$2$ .

$\frac{3 \cdot 16}{1}$

Step 1.4.2.2.2.3

Multiply

$3$ by

$16$ .

$\frac{48}{1}$

Step 1.4.2.2.2.4

Divide

$48$ by

$1$ .

$48$

Step 1.4.3

Evaluate at

$x = 2$ .

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Step 1.4.3.1

Substitute

$2$ for

$x$ .

$\frac{6 {(2)}^{2}}{(2) - 2}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Subtract

$2$ from

$2$ .

$\frac{6 {(2)}^{2}}{0}$

Step 1.4.3.2.2

The expression contains a division by

$0$ . The expression is undefined.

Undefined

Step 1.4.4

List all of the points.

$(0, 0), (4, 48)$

Step 2

Exclude the points that are not on the interval.

$(0, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at

$x = - 2$ .

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Step 3.1.1

Substitute

$- 2$ for

$x$ .

$\frac{6 {(- 2)}^{2}}{(- 2) - 2}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Cancel the common factor of

$6$ and

$(- 2) - 2$ .

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Step 3.1.2.1.1

Factor

$2$ out of

$6 {(- 2)}^{2}$ .

$\frac{2 (3 {(- 2)}^{2})}{(- 2) - 2}$

Step 3.1.2.1.2

Cancel the common factors.

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Step 3.1.2.1.2.1

Factor

$2$ out of

$- 2$ .

$\frac{2 (3 {(- 2)}^{2})}{2 (- 1) - 2}$

Step 3.1.2.1.2.2

Factor

$2$ out of

$- 2$ .

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

Step 3.1.2.1.2.3

Factor

$2$ out of

$2 \cdot - 1 + 2 \cdot - 1$ .

$\frac{2 (3 {(- 2)}^{2})}{2 \cdot (- 1 - 1)}$

Step 3.1.2.1.2.4

Cancel the common factor.

$\frac{2 (3 {(- 2)}^{2})}{2 \cdot (- 1 - 1)}$

Step 3.1.2.1.2.5

Rewrite the expression.

$\frac{3 {(- 2)}^{2}}{- 1 - 1}$

Step 3.1.2.2

Simplify the expression.

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Step 3.1.2.2.1

Raise

$- 2$ to the power of

$2$ .

$\frac{3 \cdot 4}{- 1 - 1}$

Step 3.1.2.2.2

Subtract

$1$ from

$- 1$ .

$\frac{3 \cdot 4}{- 2}$

Step 3.1.2.2.3

Multiply

$3$ by

$4$ .

$\frac{12}{- 2}$

Step 3.1.2.2.4

Divide

$12$ by

$- 2$ .

$- 6$

Step 3.2

Evaluate at

$x = 1$ .

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Step 3.2.1

Substitute

$1$ for

$x$ .

$\frac{6 {(1)}^{2}}{(1) - 2}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

One to any power is one.

$\frac{6 \cdot 1}{1 - 2}$

Step 3.2.2.2

Subtract

$2$ from

$1$ .

$\frac{6 \cdot 1}{- 1}$

Step 3.2.2.3

Multiply

$6$ by

$1$ .

$\frac{6}{- 1}$

Step 3.2.2.4

Divide

$6$ by

$- 1$ .

$- 6$

Step 3.3

List all of the points.

$(- 2, - 6), (1, - 6)$

Step 4

Compare the

$g (x)$ values found for each value of

$x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest

$g (x)$ value and the minimum will occur at the lowest

$g (x)$ value.

Absolute Maximum:

$(0, 0)$

Absolute Minimum:

$(- 2, - 6), (1, - 6)$

Step 5