Calculus Examples

$f (x) = 9 + 4 x - 4 x^{2}$ ; $[0, 5]$

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 $9 + 4 x - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4 x^{2}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.1.1.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [4 x]$ .

Tap for more steps...

Step 1.1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$0 + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 x^{2}]$

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 $n = 1$ .

$0 + 4 \cdot 1 + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.1.2.3

Multiply $4$ by $1$ .

$0 + 4 + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

Tap for more steps...

Step 1.1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$0 + 4 - 4 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$0 + 4 - 4 (2 x)$

Step 1.1.1.3.3

Multiply $2$ by $- 4$ .

$0 + 4 - 8 x$

Step 1.1.1.4

Simplify.

Tap for more steps...

Step 1.1.1.4.1

Add $0$ and $4$ .

$4 - 8 x$

Step 1.1.1.4.2

Reorder terms.

$f' (x) = - 8 x + 4$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- 8 x + 4$ .

$- 8 x + 4$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- 8 x + 4 = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$- 8 x + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$- 8 x = - 4$

Step 1.2.3

Divide each term in $- 8 x = - 4$ by $- 8$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- 8 x = - 4$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 8}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Cancel the common factor of $- 4$ and $- 8$ .

Tap for more steps...

Step 1.2.3.3.1.1

Factor $- 4$ out of $- 4$ .

$x = \frac{- 4 (1)}{- 8}$

Step 1.2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.3.1.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2}$

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 $9 + 4 x - 4 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = \frac{1}{2}$ .

Tap for more steps...

Step 1.4.1.1

Substitute $\frac{1}{2}$ for $x$ .

$9 + 4 (\frac{1}{2}) - 4 {(\frac{1}{2})}^{2}$

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

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.1.2.1.1.1

Factor $2$ out of $4$ .

$9 + 2 (2) \frac{1}{2} - 4 {(\frac{1}{2})}^{2}$

Step 1.4.1.2.1.1.2

Cancel the common factor.

$9 + 2 \cdot 2 \frac{1}{2} - 4 {(\frac{1}{2})}^{2}$

Step 1.4.1.2.1.1.3

Rewrite the expression.

$9 + 2 - 4 {(\frac{1}{2})}^{2}$

Step 1.4.1.2.1.2

Apply the product rule to $\frac{1}{2}$ .

$9 + 2 - 4 \frac{1^{2}}{2^{2}}$

Step 1.4.1.2.1.3

One to any power is one.

$9 + 2 - 4 \frac{1}{2^{2}}$

Step 1.4.1.2.1.4

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

$9 + 2 - 4 (\frac{1}{4})$

Step 1.4.1.2.1.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.4.1.2.1.5.1

Factor $4$ out of $- 4$ .

$9 + 2 + 4 (- 1) \frac{1}{4}$

Step 1.4.1.2.1.5.2

Cancel the common factor.

$9 + 2 + 4 \cdot - 1 \frac{1}{4}$

Step 1.4.1.2.1.5.3

Rewrite the expression.

$9 + 2 - 1$

Step 1.4.1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.4.1.2.2.1

Add $9$ and $2$ .

$11 - 1$

Step 1.4.1.2.2.2

Subtract $1$ from $11$ .

$10$

Step 1.4.2

List all of the points.

$(\frac{1}{2}, 10)$

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 2.1.1

Substitute $0$ for $x$ .

$9 + 4 (0) - 4 {(0)}^{2}$

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

Multiply $4$ by $0$ .

$9 + 0 - 4 {(0)}^{2}$

Step 2.1.2.1.2

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

$9 + 0 - 4 \cdot 0$

Step 2.1.2.1.3

Multiply $- 4$ by $0$ .

$9 + 0 + 0$

Step 2.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 2.1.2.2.1

Add $9$ and $0$ .

$9 + 0$

Step 2.1.2.2.2

Add $9$ and $0$ .

$9$

Step 2.2

Evaluate at $x = 5$ .

Tap for more steps...

Step 2.2.1

Substitute $5$ for $x$ .

$9 + 4 (5) - 4 {(5)}^{2}$

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

Multiply $4$ by $5$ .

$9 + 20 - 4 {(5)}^{2}$

Step 2.2.2.1.2

Raise $5$ to the power of $2$ .

$9 + 20 - 4 \cdot 25$

Step 2.2.2.1.3

Multiply $- 4$ by $25$ .

$9 + 20 - 100$

Step 2.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.2.2.1

Add $9$ and $20$ .

$29 - 100$

Step 2.2.2.2.2

Subtract $100$ from $29$ .

$- 71$

Step 2.3

List all of the points.

$(0, 9), (5, - 71)$

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: $(\frac{1}{2}, 10)$

Absolute Minimum: $(5, - 71)$

Step 4