Calculus Examples

$F (x) = x^{3} - x^{2} - 8 x + 12$ ; $[- 2, 0]$

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

Differentiate.

Tap for more steps...

Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{3} - x^{2} - 8 x + 12$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$

Step 1.1.1.1.2

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

$3 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$

Step 1.1.1.2

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

Tap for more steps...

Step 1.1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$

Step 1.1.1.2.2

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

$3 x^{2} - (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$

Step 1.1.1.2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [12]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 8 x]$ .

Tap for more steps...

Step 1.1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$3 x^{2} - 2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [12]$

Step 1.1.1.3.2

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

$3 x^{2} - 2 x - 8 \cdot 1 + \frac{d}{d x} [12]$

Step 1.1.1.3.3

Multiply $- 8$ by $1$ .

$3 x^{2} - 2 x - 8 + \frac{d}{d x} [12]$

Step 1.1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.4.1

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

$3 x^{2} - 2 x - 8 + 0$

Step 1.1.1.4.2

Add $3 x^{2} - 2 x - 8$ and $0$ .

$f' (x) = 3 x^{2} - 2 x - 8$

Step 1.1.2

The first derivative of $F (x)$ with respect to $x$ is $3 x^{2} - 2 x - 8$ .

$3 x^{2} - 2 x - 8$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 2 x - 8 = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 2 x - 8 = 0$

Step 1.2.2

Factor by grouping.

Tap for more steps...

Step 1.2.2.1

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 2$ .

Tap for more steps...

Step 1.2.2.1.1

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

$3 x^{2} - 2 x - 8 = 0$

Step 1.2.2.1.2

Rewrite $- 2$ as $4$ plus $- 6$

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

Step 1.2.2.1.3

Apply the distributive property.

$3 x^{2} + 4 x - 6 x - 8 = 0$

Step 1.2.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2.2

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

$x (3 x + 4) - 2 (3 x + 4) = 0$

Step 1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$(3 x + 4) (x - 2) = 0$

Step 1.2.3

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

$3 x + 4 = 0$

$x - 2 = 0$

Step 1.2.4

Set $3 x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 1.2.4.2

Solve $3 x + 4 = 0$ for $x$ .

Tap for more steps...

Step 1.2.4.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 1.2.4.2.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

Tap for more steps...

Step 1.2.4.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 1.2.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.4.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 1.2.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$x - 2 = 0$

Step 1.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.6

The final solution is all the values that make $(3 x + 4) (x - 2) = 0$ true.

$x = - \frac{4}{3}, 2$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

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 1.4

Evaluate $x^{3} - x^{2} - 8 x + 12$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = - \frac{4}{3}$ .

Tap for more steps...

Step 1.4.1.1

Substitute $- \frac{4}{3}$ for $x$ .

${(- \frac{4}{3})}^{3} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1.1

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

Tap for more steps...

Step 1.4.1.2.1.1.1

Apply the product rule to $- \frac{4}{3}$ .

${(- 1)}^{3} {(\frac{4}{3})}^{3} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.1.2

Apply the product rule to $\frac{4}{3}$ .

${(- 1)}^{3} \frac{4^{3}}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.2

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

$- \frac{4^{3}}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.3

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

$- \frac{64}{3^{3}} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.4

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

$- \frac{64}{27} - {(- \frac{4}{3})}^{2} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.5

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

Tap for more steps...

Step 1.4.1.2.1.5.1

Apply the product rule to $- \frac{4}{3}$ .

$- \frac{64}{27} - ({(- 1)}^{2} {(\frac{4}{3})}^{2}) - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.5.2

Apply the product rule to $\frac{4}{3}$ .

$- \frac{64}{27} - ({(- 1)}^{2} \frac{4^{2}}{3^{2}}) - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 1.4.1.2.1.6.1

Move ${(- 1)}^{2}$ .

$- \frac{64}{27} + {(- 1)}^{2} \cdot - 1 \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.6.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

Tap for more steps...

Step 1.4.1.2.1.6.2.1

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

$- \frac{64}{27} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.6.2.2

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

$- \frac{64}{27} + {(- 1)}^{2 + 1} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.6.3

Add $2$ and $1$ .

$- \frac{64}{27} + {(- 1)}^{3} \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.7

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

$- \frac{64}{27} - \frac{4^{2}}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.8

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

$- \frac{64}{27} - \frac{16}{3^{2}} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.9

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

$- \frac{64}{27} - \frac{16}{9} - 8 (- \frac{4}{3}) + 12$

Step 1.4.1.2.1.10

Multiply $- 8 (- \frac{4}{3})$ .

Tap for more steps...

Step 1.4.1.2.1.10.1

Multiply $- 1$ by $- 8$ .

$- \frac{64}{27} - \frac{16}{9} + 8 (\frac{4}{3}) + 12$

Step 1.4.1.2.1.10.2

Combine $8$ and $\frac{4}{3}$ .

$- \frac{64}{27} - \frac{16}{9} + \frac{8 \cdot 4}{3} + 12$

Step 1.4.1.2.1.10.3

Multiply $8$ by $4$ .

$- \frac{64}{27} - \frac{16}{9} + \frac{32}{3} + 12$

Step 1.4.1.2.2

Find the common denominator.

Tap for more steps...

Step 1.4.1.2.2.1

Multiply $\frac{16}{9}$ by $\frac{3}{3}$ .

$- \frac{64}{27} - (\frac{16}{9} \cdot \frac{3}{3}) + \frac{32}{3} + 12$

Step 1.4.1.2.2.2

Multiply $\frac{16}{9}$ by $\frac{3}{3}$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32}{3} + 12$

Step 1.4.1.2.2.3

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32}{3} \cdot \frac{9}{9} + 12$

Step 1.4.1.2.2.4

Multiply $\frac{32}{3}$ by $\frac{9}{9}$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + 12$

Step 1.4.1.2.2.5

Write $12$ as a fraction with denominator $1$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{12}{1}$

Step 1.4.1.2.2.6

Multiply $\frac{12}{1}$ by $\frac{27}{27}$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{12}{1} \cdot \frac{27}{27}$

Step 1.4.1.2.2.7

Multiply $\frac{12}{1}$ by $\frac{27}{27}$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{9 \cdot 3} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{12 \cdot 27}{27}$

Step 1.4.1.2.2.8

Reorder the factors of $9 \cdot 3$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{3 \cdot 9} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{12 \cdot 27}{27}$

Step 1.4.1.2.2.9

Multiply $3$ by $9$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{27} + \frac{32 \cdot 9}{3 \cdot 9} + \frac{12 \cdot 27}{27}$

Step 1.4.1.2.2.10

Multiply $3$ by $9$ .

$- \frac{64}{27} - \frac{16 \cdot 3}{27} + \frac{32 \cdot 9}{27} + \frac{12 \cdot 27}{27}$

Step 1.4.1.2.3

Combine the numerators over the common denominator.

$\frac{- 64 - 16 \cdot 3 + 32 \cdot 9 + 12 \cdot 27}{27}$

Step 1.4.1.2.4

Simplify each term.

Tap for more steps...

Step 1.4.1.2.4.1

Multiply $- 16$ by $3$ .

$\frac{- 64 - 48 + 32 \cdot 9 + 12 \cdot 27}{27}$

Step 1.4.1.2.4.2

Multiply $32$ by $9$ .

$\frac{- 64 - 48 + 288 + 12 \cdot 27}{27}$

Step 1.4.1.2.4.3

Multiply $12$ by $27$ .

$\frac{- 64 - 48 + 288 + 324}{27}$

Step 1.4.1.2.5

Simplify by adding and subtracting.

Tap for more steps...

Step 1.4.1.2.5.1

Subtract $48$ from $- 64$ .

$\frac{- 112 + 288 + 324}{27}$

Step 1.4.1.2.5.2

Add $- 112$ and $288$ .

$\frac{176 + 324}{27}$

Step 1.4.1.2.5.3

Add $176$ and $324$ .

$\frac{500}{27}$

Step 1.4.2

Evaluate at $x = 2$ .

Tap for more steps...

Step 1.4.2.1

Substitute $2$ for $x$ .

${(2)}^{3} - {(2)}^{2} - 8 \cdot 2 + 12$

Step 1.4.2.2

Simplify.

Tap for more steps...

Step 1.4.2.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.2.1.1

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

$8 - {(2)}^{2} - 8 \cdot 2 + 12$

Step 1.4.2.2.1.2

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

$8 - 1 \cdot 4 - 8 \cdot 2 + 12$

Step 1.4.2.2.1.3

Multiply $- 1$ by $4$ .

$8 - 4 - 8 \cdot 2 + 12$

Step 1.4.2.2.1.4

Multiply $- 8$ by $2$ .

$8 - 4 - 16 + 12$

Step 1.4.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.4.2.2.2.1

Subtract $4$ from $8$ .

$4 - 16 + 12$

Step 1.4.2.2.2.2

Subtract $16$ from $4$ .

$- 12 + 12$

Step 1.4.2.2.2.3

Add $- 12$ and $12$ .

$0$

Step 1.4.3

List all of the points.

$(- \frac{4}{3}, \frac{500}{27}), (2, 0)$

Step 2

Exclude the points that are not on the interval.

$(- \frac{4}{3}, \frac{500}{27})$

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 $x$ .

${(- 2)}^{3} - {(- 2)}^{2} - 8 \cdot - 2 + 12$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

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

$- 8 - {(- 2)}^{2} - 8 \cdot - 2 + 12$

Step 3.1.2.1.2

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

$- 8 - 1 \cdot 4 - 8 \cdot - 2 + 12$

Step 3.1.2.1.3

Multiply $- 1$ by $4$ .

$- 8 - 4 - 8 \cdot - 2 + 12$

Step 3.1.2.1.4

Multiply $- 8$ by $- 2$ .

$- 8 - 4 + 16 + 12$

Step 3.1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.1.2.2.1

Subtract $4$ from $- 8$ .

$- 12 + 16 + 12$

Step 3.1.2.2.2

Add $- 12$ and $16$ .

$4 + 12$

Step 3.1.2.2.3

Add $4$ and $12$ .

$16$

Step 3.2

Evaluate at $x = 0$ .

Tap for more steps...

Step 3.2.1

Substitute $0$ for $x$ .

${(0)}^{3} - {(0)}^{2} - 8 \cdot 0 + 12$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Raising $0$ to any positive power yields $0$ .

$0 - {(0)}^{2} - 8 \cdot 0 + 12$

Step 3.2.2.1.2

Raising $0$ to any positive power yields $0$ .

$0 - 0 - 8 \cdot 0 + 12$

Step 3.2.2.1.3

Multiply $- 1$ by $0$ .

$0 + 0 - 8 \cdot 0 + 12$

Step 3.2.2.1.4

Multiply $- 8$ by $0$ .

$0 + 0 + 0 + 12$

Step 3.2.2.2

Simplify by adding numbers.

Tap for more steps...

Step 3.2.2.2.1

Add $0$ and $0$ .

$0 + 0 + 12$

Step 3.2.2.2.2

Add $0$ and $0$ .

$0 + 12$

Step 3.2.2.2.3

Add $0$ and $12$ .

$12$

Step 3.3

List all of the points.

$(- 2, 16), (0, 12)$

Step 4

Compare the $F (x)$ 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 $F (x)$ value and the minimum will occur at the lowest $F (x)$ value.

Absolute Maximum: $(- \frac{4}{3}, \frac{500}{27})$

Absolute Minimum: $(0, 12)$

Step 5