Calculus Examples

$f (x) = x^{4} + 8 x^{3} + 18 x^{2} + 4$ , $[- 4, 1]$

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

Differentiate.

Tap for more steps...

Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{4} + 8 x^{3} + 18 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$

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 $n = 4$ .

$4 x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

Tap for more steps...

Step 1.1.1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} + 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$

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 $n = 3$ .

$4 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$

Step 1.1.1.2.3

Multiply $3$ by $8$ .

$4 x^{3} + 24 x^{2} + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [4]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

Tap for more steps...

Step 1.1.1.3.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} + 24 x^{2} + 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

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 $n = 2$ .

$4 x^{3} + 24 x^{2} + 18 (2 x) + \frac{d}{d x} [4]$

Step 1.1.1.3.3

Multiply $2$ by $18$ .

$4 x^{3} + 24 x^{2} + 36 x + \frac{d}{d x} [4]$

Step 1.1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$4 x^{3} + 24 x^{2} + 36 x + 0$

Step 1.1.1.4.2

Add $4 x^{3} + 24 x^{2} + 36 x$ and $0$ .

$f' (x) = 4 x^{3} + 24 x^{2} + 36 x$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} + 24 x^{2} + 36 x$ .

$4 x^{3} + 24 x^{2} + 36 x$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} + 24 x^{2} + 36 x = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$4 x^{3} + 24 x^{2} + 36 x = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $4 x$ out of $4 x^{3} + 24 x^{2} + 36 x$ .

Tap for more steps...

Step 1.2.2.1.1

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

$4 x (x^{2}) + 24 x^{2} + 36 x = 0$

Step 1.2.2.1.2

Factor $4 x$ out of $24 x^{2}$ .

$4 x (x^{2}) + 4 x (6 x) + 36 x = 0$

Step 1.2.2.1.3

Factor $4 x$ out of $36 x$ .

$4 x (x^{2}) + 4 x (6 x) + 4 x (9) = 0$

Step 1.2.2.1.4

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

$4 x (x^{2} + 6 x) + 4 x (9) = 0$

Step 1.2.2.1.5

Factor $4 x$ out of $4 x (x^{2} + 6 x) + 4 x (9)$ .

$4 x (x^{2} + 6 x + 9) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.2.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.2.2.2

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

$6 x = 2 \cdot x \cdot 3$

Step 1.2.2.2.3

Rewrite the polynomial.

$4 x (x^{2} + 2 \cdot x \cdot 3 + 3^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

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 $0$ .

$x = 0$

${(x + 3)}^{2} = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set ${(x + 3)}^{2}$ equal to $0$ .

${(x + 3)}^{2} = 0$

Step 1.2.5.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

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

$x + 3 = 0$

Step 1.2.5.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

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

$x = 0, - 3$

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 $x^{4} + 8 x^{3} + 18 x^{2} + 4$ 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 $x$ .

${(0)}^{4} + 8 {(0)}^{3} + 18 {(0)}^{2} + 4$

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

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

$0 + 8 {(0)}^{3} + 18 {(0)}^{2} + 4$

Step 1.4.1.2.1.2

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

$0 + 8 \cdot 0 + 18 {(0)}^{2} + 4$

Step 1.4.1.2.1.3

Multiply $8$ by $0$ .

$0 + 0 + 18 {(0)}^{2} + 4$

Step 1.4.1.2.1.4

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

$0 + 0 + 18 \cdot 0 + 4$

Step 1.4.1.2.1.5

Multiply $18$ by $0$ .

$0 + 0 + 0 + 4$

Step 1.4.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 1.4.1.2.2.1

Add $0$ and $0$ .

$0 + 0 + 4$

Step 1.4.1.2.2.2

Add $0$ and $0$ .

$0 + 4$

Step 1.4.1.2.2.3

Add $0$ and $4$ .

$4$

Step 1.4.2

Evaluate at $x = - 3$ .

Tap for more steps...

Step 1.4.2.1

Substitute $- 3$ for $x$ .

${(- 3)}^{4} + 8 {(- 3)}^{3} + 18 {(- 3)}^{2} + 4$

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

$81 + 8 {(- 3)}^{3} + 18 {(- 3)}^{2} + 4$

Step 1.4.2.2.1.2

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

$81 + 8 \cdot - 27 + 18 {(- 3)}^{2} + 4$

Step 1.4.2.2.1.3

Multiply $8$ by $- 27$ .

$81 - 216 + 18 {(- 3)}^{2} + 4$

Step 1.4.2.2.1.4

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

$81 - 216 + 18 \cdot 9 + 4$

Step 1.4.2.2.1.5

Multiply $18$ by $9$ .

$81 - 216 + 162 + 4$

Step 1.4.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.4.2.2.2.1

Subtract $216$ from $81$ .

$- 135 + 162 + 4$

Step 1.4.2.2.2.2

Add $- 135$ and $162$ .

$27 + 4$

Step 1.4.2.2.2.3

Add $27$ and $4$ .

$31$

Step 1.4.3

List all of the points.

$(0, 4), (- 3, 31)$

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $x = - 4$ .

Tap for more steps...

Step 2.1.1

Substitute $- 4$ for $x$ .

${(- 4)}^{4} + 8 {(- 4)}^{3} + 18 {(- 4)}^{2} + 4$

Step 2.1.2

Simplify.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

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

$256 + 8 {(- 4)}^{3} + 18 {(- 4)}^{2} + 4$

Step 2.1.2.1.2

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

$256 + 8 \cdot - 64 + 18 {(- 4)}^{2} + 4$

Step 2.1.2.1.3

Multiply $8$ by $- 64$ .

$256 - 512 + 18 {(- 4)}^{2} + 4$

Step 2.1.2.1.4

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

$256 - 512 + 18 \cdot 16 + 4$

Step 2.1.2.1.5

Multiply $18$ by $16$ .

$256 - 512 + 288 + 4$

Step 2.1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.1.2.2.1

Subtract $512$ from $256$ .

$- 256 + 288 + 4$

Step 2.1.2.2.2

Add $- 256$ and $288$ .

$32 + 4$

Step 2.1.2.2.3

Add $32$ and $4$ .

$36$

Step 2.2

Evaluate at $x = 1$ .

Tap for more steps...

Step 2.2.1

Substitute $1$ for $x$ .

${(1)}^{4} + 8 {(1)}^{3} + 18 {(1)}^{2} + 4$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

One to any power is one.

$1 + 8 {(1)}^{3} + 18 {(1)}^{2} + 4$

Step 2.2.2.1.2

One to any power is one.

$1 + 8 \cdot 1 + 18 {(1)}^{2} + 4$

Step 2.2.2.1.3

Multiply $8$ by $1$ .

$1 + 8 + 18 {(1)}^{2} + 4$

Step 2.2.2.1.4

One to any power is one.

$1 + 8 + 18 \cdot 1 + 4$

Step 2.2.2.1.5

Multiply $18$ by $1$ .

$1 + 8 + 18 + 4$

Step 2.2.2.2

Simplify by adding numbers.

Tap for more steps...

Step 2.2.2.2.1

Add $1$ and $8$ .

$9 + 18 + 4$

Step 2.2.2.2.2

Add $9$ and $18$ .

$27 + 4$

Step 2.2.2.2.3

Add $27$ and $4$ .

$31$

Step 2.3

List all of the points.

$(- 4, 36), (1, 31)$

Step 3

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: $(- 4, 36)$

Absolute Minimum: $(0, 4)$

Step 4