Calculus Examples

$g (x) = \frac{6 x^{2}}{x - 2}$ , $[- 2, 1]$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x - 2$ .

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

Step 1.1.1.3

Differentiate.

Tap for more steps...

Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Move $2$ to the left of $x - 2$ .

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

Step 1.1.1.3.3

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

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

Step 1.1.1.3.4

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

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

Step 1.1.1.3.5

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

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

Step 1.1.1.3.6

Combine fractions.

Tap for more steps...

Step 1.1.1.3.6.1

Add $1$ and $0$ .

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

Step 1.1.1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.1.1.3.6.3

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

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

Step 1.1.1.4

Simplify.

Tap for more steps...

Step 1.1.1.4.1

Apply the distributive property.

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

Step 1.1.1.4.2

Apply the distributive property.

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

Step 1.1.1.4.3

Apply the distributive property.

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

Step 1.1.1.4.4

Simplify the numerator.

Tap for more steps...

Step 1.1.1.4.4.1

Simplify each term.

Tap for more steps...

Step 1.1.1.4.4.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.1.4.4.1.1.1

Move $x$ .

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

Step 1.1.1.4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.1.4.4.1.2

Multiply $2$ by $6$ .

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

Step 1.1.1.4.4.1.3

Multiply $2$ by $- 2$ .

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

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

Tap for more steps...

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

Tap for more steps...

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 $x$ .

Tap for more steps...

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 $x$ .

Tap for more steps...

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.

Tap for more steps...

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 $x$ .

Tap for more steps...

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.

Tap for more steps...

Step 1.4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 1.4.1.1

Substitute $0$ for $x$ .

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

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Cancel the common factor of $6$ and $(0) - 2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $x = 4$ .

Tap for more steps...

Step 1.4.2.1

Substitute $4$ for $x$ .

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

Step 1.4.2.2

Simplify.

Tap for more steps...

Step 1.4.2.2.1

Cancel the common factor of $6$ and $(4) - 2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $x = 2$ .

Tap for more steps...

Step 1.4.3.1

Substitute $2$ for $x$ .

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

Step 1.4.3.2

Simplify.

Tap for more steps...

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.

Tap for more steps...

Step 3.1

Evaluate at $x = - 2$ .

Tap for more steps...

Step 3.1.1

Substitute $- 2$ for $x$ .

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

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Cancel the common factor of $6$ and $(- 2) - 2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $x = 1$ .

Tap for more steps...

Step 3.2.1

Substitute $1$ for $x$ .

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

Step 3.2.2

Simplify.

Tap for more steps...

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