Calculus Examples

$f (x) = \frac{2 x + 5}{3}$ , $[0, 5]$

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

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x + 5}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [2 x + 5]$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Multiply $2$ by $1$ .

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

Step 1.1.1.6

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{3} (2 + 0)$

Step 1.1.1.7

Combine fractions.

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Step 1.1.1.7.1

Add $2$ and $0$ .

$\frac{1}{3} \cdot 2$

Step 1.1.1.7.2

Combine $\frac{1}{3}$ and $2$ .

$f' (x) = \frac{2}{3}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{3} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{2}{3} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 = 0$

Step 1.2.3

Since $2 \neq 0$ , 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 $x$ .

$\frac{2 (0) + 5}{3}$

Step 2.1.2

Simplify the numerator.

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Step 2.1.2.1

Multiply $2$ by $0$ .

$\frac{0 + 5}{3}$

Step 2.1.2.2

Add $0$ and $5$ .

$\frac{5}{3}$

Step 2.2

Evaluate at $x = 5$ .

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Step 2.2.1

Substitute $5$ for $x$ .

$\frac{2 (5) + 5}{3}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

Multiply $2$ by $5$ .

$\frac{10 + 5}{3}$

Step 2.2.2.1.2

Add $10$ and $5$ .

$\frac{15}{3}$

Step 2.2.2.2

Divide $15$ by $3$ .

$5$

Step 2.3

List all of the points.

$(0, \frac{5}{3}), (5, 5)$

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: $(5, 5)$

Absolute Minimum: $(0, \frac{5}{3})$

Step 4