Calculus Examples

$f (x) = | x |$ , given

$- 8 \leq x \leq 7$

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

The derivative of

$| x |$ with respect to

$x$ is

$\frac{x}{| x |}$ .

$f' (x) = \frac{x}{| x |}$

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{x}{| x |}$ .

$\frac{x}{| x |}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{x}{| x |} = 0$ .

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Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{x}{| x |} = 0$

Step 1.2.2

Set the numerator equal to zero.

$x = 0$

Step 1.2.3

Exclude the solutions that do not make

$\frac{x}{| x |} = 0$ true.

$No solution$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in

$\frac{x}{| x |}$ equal to

$0$ to find where the expression is undefined.

$| x | = 0$

Step 1.3.2

Solve for

$x$ .

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Step 1.3.2.1

Remove the absolute value term. This creates a

$\pm$ on the right side of the equation because

$| x | = \pm x$ .

$x = \pm 0$

Step 1.3.2.2

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 1.4

Evaluate

$| x |$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 0$ .

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Step 1.4.1.1

Substitute

$0$ for

$x$ .

$| 0 |$

Step 1.4.1.2

The absolute value is the distance between a number and zero. The distance between

$0$ and

$0$ is

$0$ .

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at

$x = - 8$ .

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Step 2.1.1

Substitute

$- 8$ for

$x$ .

$| - 8 |$

Step 2.1.2

The absolute value is the distance between a number and zero. The distance between

$- 8$ and

$0$ is

$8$ .

$8$

Step 2.2

Evaluate at

$x = 7$ .

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Step 2.2.1

Substitute

$7$ for

$x$ .

$| 7 |$

Step 2.2.2

The absolute value is the distance between a number and zero. The distance between

$0$ and

$7$ is

$7$ .

$7$

Step 2.3

List all of the points.

$(- 8, 8), (7, 7)$

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:

$(- 8, 8)$

Absolute Minimum:

$(0, 0)$

Step 4