Calculus Examples

$f (x) = 5 x^{2} - 3 x - 1$ , $[- 1, 3]$

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

By the Sum Rule, the derivative of $5 x^{2} - 3 x - 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 1]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

Tap for more steps...

Step 1.1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 1]$

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 $n = 2$ .

$5 (2 x) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 1]$

Step 1.1.1.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 1]$

Step 1.1.1.3

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

Tap for more steps...

Step 1.1.1.3.1

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

$10 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$10 x - 3 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.1.3.3

Multiply $- 3$ by $1$ .

$10 x - 3 + \frac{d}{d x} [- 1]$

Step 1.1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$10 x - 3 + 0$

Step 1.1.1.4.2

Add $10 x - 3$ and $0$ .

$f' (x) = 10 x - 3$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $10 x - 3$ .

$10 x - 3$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $10 x - 3 = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$10 x - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$10 x = 3$

Step 1.2.3

Divide each term in $10 x = 3$ by $10$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $10 x = 3$ by $10$ .

$\frac{10 x}{10} = \frac{3}{10}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{3}{10}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{10}$

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 $5 x^{2} - 3 x - 1$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = \frac{3}{10}$ .

Tap for more steps...

Step 1.4.1.1

Substitute $\frac{3}{10}$ for $x$ .

$5 {(\frac{3}{10})}^{2} - 3 (\frac{3}{10}) - 1$

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

Apply the product rule to $\frac{3}{10}$ .

$5 \frac{3^{2}}{10^{2}} - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.2

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

$5 \frac{9}{10^{2}} - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.3

Raise $10$ to the power of $2$ .

$5 (\frac{9}{100}) - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.4.1.2.1.4.1

Factor $5$ out of $100$ .

$5 \frac{9}{5 (20)} - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.4.2

Cancel the common factor.

$5 \frac{9}{5 \cdot 20} - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.4.3

Rewrite the expression.

$\frac{9}{20} - 3 (\frac{3}{10}) - 1$

Step 1.4.1.2.1.5

Multiply $- 3 (\frac{3}{10})$ .

Tap for more steps...

Step 1.4.1.2.1.5.1

Combine $- 3$ and $\frac{3}{10}$ .

$\frac{9}{20} + \frac{- 3 \cdot 3}{10} - 1$

Step 1.4.1.2.1.5.2

Multiply $- 3$ by $3$ .

$\frac{9}{20} + \frac{- 9}{10} - 1$

Step 1.4.1.2.1.6

Move the negative in front of the fraction.

$\frac{9}{20} - \frac{9}{10} - 1$

Step 1.4.1.2.2

Find the common denominator.

Tap for more steps...

Step 1.4.1.2.2.1

Multiply $\frac{9}{10}$ by $\frac{2}{2}$ .

$\frac{9}{20} - (\frac{9}{10} \cdot \frac{2}{2}) - 1$

Step 1.4.1.2.2.2

Multiply $\frac{9}{10}$ by $\frac{2}{2}$ .

$\frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} - 1$

Step 1.4.1.2.2.3

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

$\frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 1}{1}$

Step 1.4.1.2.2.4

Multiply $\frac{- 1}{1}$ by $\frac{20}{20}$ .

$\frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 1}{1} \cdot \frac{20}{20}$

Step 1.4.1.2.2.5

Multiply $\frac{- 1}{1}$ by $\frac{20}{20}$ .

$\frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 1 \cdot 20}{20}$

Step 1.4.1.2.2.6

Reorder the factors of $10 \cdot 2$ .

$\frac{9}{20} - \frac{9 \cdot 2}{2 \cdot 10} + \frac{- 1 \cdot 20}{20}$

Step 1.4.1.2.2.7

Multiply $2$ by $10$ .

$\frac{9}{20} - \frac{9 \cdot 2}{20} + \frac{- 1 \cdot 20}{20}$

Step 1.4.1.2.3

Combine the numerators over the common denominator.

$\frac{9 - 9 \cdot 2 - 1 \cdot 20}{20}$

Step 1.4.1.2.4

Simplify each term.

Tap for more steps...

Step 1.4.1.2.4.1

Multiply $- 9$ by $2$ .

$\frac{9 - 18 - 1 \cdot 20}{20}$

Step 1.4.1.2.4.2

Multiply $- 1$ by $20$ .

$\frac{9 - 18 - 20}{20}$

Step 1.4.1.2.5

Simplify the expression.

Tap for more steps...

Step 1.4.1.2.5.1

Subtract $18$ from $9$ .

$\frac{- 9 - 20}{20}$

Step 1.4.1.2.5.2

Subtract $20$ from $- 9$ .

$\frac{- 29}{20}$

Step 1.4.1.2.5.3

Move the negative in front of the fraction.

$- \frac{29}{20}$

Step 1.4.2

List all of the points.

$(\frac{3}{10}, - \frac{29}{20})$

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $x = - 1$ .

Tap for more steps...

Step 2.1.1

Substitute $- 1$ for $x$ .

$5 {(- 1)}^{2} - 3 \cdot - 1 - 1$

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 $- 1$ to the power of $2$ .

$5 \cdot 1 - 3 \cdot - 1 - 1$

Step 2.1.2.1.2

Multiply $5$ by $1$ .

$5 - 3 \cdot - 1 - 1$

Step 2.1.2.1.3

Multiply $- 3$ by $- 1$ .

$5 + 3 - 1$

Step 2.1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.1.2.2.1

Add $5$ and $3$ .

$8 - 1$

Step 2.1.2.2.2

Subtract $1$ from $8$ .

$7$

Step 2.2

Evaluate at $x = 3$ .

Tap for more steps...

Step 2.2.1

Substitute $3$ for $x$ .

$5 {(3)}^{2} - 3 \cdot 3 - 1$

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

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

$5 \cdot 9 - 3 \cdot 3 - 1$

Step 2.2.2.1.2

Multiply $5$ by $9$ .

$45 - 3 \cdot 3 - 1$

Step 2.2.2.1.3

Multiply $- 3$ by $3$ .

$45 - 9 - 1$

Step 2.2.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 2.2.2.2.1

Subtract $9$ from $45$ .

$36 - 1$

Step 2.2.2.2.2

Subtract $1$ from $36$ .

$35$

Step 2.3

List all of the points.

$(- 1, 7), (3, 35)$

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: $(3, 35)$

Absolute Minimum: $(\frac{3}{10}, - \frac{29}{20})$

Step 4