Calculus Examples

$f (x) = \frac{3}{4} x + 2$ , $[0, 4]$

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 $\frac{3}{4} x + 2$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{4} x] + \frac{d}{d x} [2]$ .

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3}{4} x$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [x]$ .

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Multiply $\frac{3}{4}$ by $1$ .

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.1.1.3.2

Add $\frac{3}{4}$ and $0$ .

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

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