Calculus Examples

f (x) = \frac{3}{4} x + 2

$f (x) = \frac{3}{4} x + 2$ ,

[0, 4]

$[0, 4]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of

\frac{3}{4} x + 2

$\frac{3}{4} x + 2$ with respect to

x

$x$ is

\frac{d}{d x} [\frac{3}{4} x] + \frac{d}{d x} [2]

$\frac{d}{d x} [\frac{3}{4} x] + \frac{d}{d x} [2]$ .

\frac{d}{d x} [\frac{3}{4} x] + \frac{d}{d x} [2]

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

Step 1.1.1.2

Evaluate

\frac{d}{d x} [\frac{3}{4} x]

$\frac{d}{d x} [\frac{3}{4} x]$ .

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Step 1.1.1.2.1

Since

\frac{3}{4}

$\frac{3}{4}$ is constant with respect to

x

$x$ , the derivative of

\frac{3}{4} x

$\frac{3}{4} x$ with respect to

x

$x$ is

\frac{3}{4} \frac{d}{d x} [x]

$\frac{3}{4} \frac{d}{d x} [x]$ .

\frac{3}{4} \frac{d}{d x} [x] + \frac{d}{d x} [2]

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

Step 1.1.1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

\frac{3}{4} \cdot 1 + \frac{d}{d x} [2]

$\frac{3}{4} \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.1.2.3

Multiply

\frac{3}{4}

$\frac{3}{4}$ by

1

$1$ .

\frac{3}{4} + \frac{d}{d x} [2]

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

\frac{3}{4} + \frac{d}{d x} [2]

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2

$2$ with respect to

x

$x$ is

0

$0$ .

\frac{3}{4} + 0

$\frac{3}{4} + 0$

Step 1.1.1.3.2

Add

\frac{3}{4}

$\frac{3}{4}$ and

0

$0$ .

$f' (x) = \frac{3}{4}$

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{3}{4}$ .

$\frac{3}{4}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{3}{4} = 0$ .

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Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{3}{4} = 0$

Step 1.2.2

Set the numerator equal to zero.

$3 = 0$

Step 1.2.3

Since

$3 \neq 0$ , there are no solutions.

No solution

Step 1.3

There are no values of

$x$ in the domain of the original problem where the derivative is

$0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at

$x = 0$ .

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Step 2.1.1

Substitute

$0$ for

$x$ .

$\frac{3}{4} \cdot (0) + 2$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Multiply

$\frac{3}{4}$ by

$0$ .

$0 + 2$

Step 2.1.2.2

Add

$0$ and

$2$ .

$2$

Step 2.2

Evaluate at

$x = 4$ .

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Step 2.2.1

Substitute

$4$ for

$x$ .

$\frac{3}{4} \cdot (4) + 2$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of

$4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3}{4} \cdot 4 + 2$

Step 2.2.2.1.2

Rewrite the expression.

$3 + 2$

Step 2.2.2.2

Add

$3$ and

$2$ .

$5$

Step 2.3

List all of the points.

$(0, 2), (4, 5)$

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:

$(4, 5)$

Absolute Minimum:

$(0, 2)$

Step 4