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OBJECTIVES:

1. Determine intervals on which a function is increasing or decreasing.

2. Apply the First Derivative Test to find the relative extrema of a function.

1. Determine intervals on which a function is increasing or decreasing.

2. Apply the First Derivative Test to find the relative extrema of a function.

1.
Applications of Differentiation

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

2.
Increasing and Decreasing

Functions and the First

Derivative Test

Copyright © Cengage Learning. All rights reserved.

Functions and the First

Derivative Test

Copyright © Cengage Learning. All rights reserved.

3.
Determine intervals on which a function is increasing or

decreasing.

Apply the First Derivative Test to find relative extrema of

a function.

3

decreasing.

Apply the First Derivative Test to find relative extrema of

a function.

3

4.
Increasing and Decreasing

Functions

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Functions

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5.
Increasing and Decreasing Functions

You will learn how derivatives can be used to classify relative

extrema as either relative minima or relative maxima. First, it

is important to define increasing and decreasing functions.

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You will learn how derivatives can be used to classify relative

extrema as either relative minima or relative maxima. First, it

is important to define increasing and decreasing functions.

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6.
Increasing and Decreasing Functions

A function is increasing when, as x moves to the right, its

graph moves up, and is decreasing when its graph moves

down. For example, the function in Figure 3.15 is

decreasing on the interval is constant on the

interval (a, b) and is increasing on the interval

Figure 3.15 6

A function is increasing when, as x moves to the right, its

graph moves up, and is decreasing when its graph moves

down. For example, the function in Figure 3.15 is

decreasing on the interval is constant on the

interval (a, b) and is increasing on the interval

Figure 3.15 6

7.
Increasing and Decreasing Functions

As shown in Theorem 3.5 below, a positive derivative

implies that the function is increasing, a negative derivative

implies that the function is decreasing, and a zero

derivative on an entire interval implies that the function is

constant on that interval.

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As shown in Theorem 3.5 below, a positive derivative

implies that the function is increasing, a negative derivative

implies that the function is decreasing, and a zero

derivative on an entire interval implies that the function is

constant on that interval.

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8.
Example 1 – Intervals on Which f Is Increasing or Decreasing

Find the open intervals on which is

increasing or decreasing.

Note that f is differentiable on the entire real number line

and the derivative of f is

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Find the open intervals on which is

increasing or decreasing.

Note that f is differentiable on the entire real number line

and the derivative of f is

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9.
Example 1 – Solution cont’d

To determine the critical numbers of f, set f'(x) equal to

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To determine the critical numbers of f, set f'(x) equal to

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10.
Example 1 – Solution cont’d

Because there are no points for which f' does not exist,

you can conclude that x = 0 and x = 1 are the only critical

numbers. The table summarizes the testing of the three

intervals determined by these two critical numbers.

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Because there are no points for which f' does not exist,

you can conclude that x = 0 and x = 1 are the only critical

numbers. The table summarizes the testing of the three

intervals determined by these two critical numbers.

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11.
Example 1 – Solution cont’d

By Theorem 3.5, f is increasing on the intervals and

and decreasing on the interval (0,1), as shown in

Figure 3.16.

Figure 3.16

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By Theorem 3.5, f is increasing on the intervals and

and decreasing on the interval (0,1), as shown in

Figure 3.16.

Figure 3.16

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12.
Increasing and Decreasing Functions

Example 1 gives you one instance of how to find intervals on

which a function is increasing or decreasing. The guidelines

below summarize the steps followed in that example.

12

Example 1 gives you one instance of how to find intervals on

which a function is increasing or decreasing. The guidelines

below summarize the steps followed in that example.

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13.
Increasing and Decreasing Functions

A function is strictly monotonic on an interval when it is

either increasing on the entire interval or decreasing on the

entire interval. For instance, the function f(x) = x3

is strictly monotonic on the entire real number line because

it is increasing on the entire real number line, as shown in

Figure 3.17(a).

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Figure 3.17(a)

A function is strictly monotonic on an interval when it is

either increasing on the entire interval or decreasing on the

entire interval. For instance, the function f(x) = x3

is strictly monotonic on the entire real number line because

it is increasing on the entire real number line, as shown in

Figure 3.17(a).

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Figure 3.17(a)

14.
Increasing and Decreasing Functions

The function shown in Figure 3.17(b) is not strictly

monotonic on the entire real number line because it is

constant on the interval [0, 1].

Figure 3.17(b)

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The function shown in Figure 3.17(b) is not strictly

monotonic on the entire real number line because it is

constant on the interval [0, 1].

Figure 3.17(b)

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15.
The First Derivative Test

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16.
The First Derivative Test

After you have determined the intervals on which a function is

increasing or decreasing, it is not difficult to locate the relative

extrema of the function.

For instance, in Figure 3.18

(from Example 1), the function

has a relative maximum at the point

(0, 0) because f is increasing

immediately to the left of x = 0

and decreasing immediately to the

right of x = 0.

Figure 3.18

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After you have determined the intervals on which a function is

increasing or decreasing, it is not difficult to locate the relative

extrema of the function.

For instance, in Figure 3.18

(from Example 1), the function

has a relative maximum at the point

(0, 0) because f is increasing

immediately to the left of x = 0

and decreasing immediately to the

right of x = 0.

Figure 3.18

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17.
The First Derivative Test

Similarly, f has a relative minimum at the point

because f is decreasing immediately to the left of x = 1 and

increasing immediately to the right of x = 1. The next

theorem makes this more explicit.

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Similarly, f has a relative minimum at the point

because f is decreasing immediately to the left of x = 1 and

increasing immediately to the right of x = 1. The next

theorem makes this more explicit.

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18.
The First Derivative Test

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19.
Example 2 – Applying the First Derivative Test

Find the relative extrema of the function

in the interval (0, 2π).

Note that f is continuous on the interval (0, 2π). The

derivative of f is

To determine the critical numbers of f in this interval, set

f'(x) equal to zero.

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Find the relative extrema of the function

in the interval (0, 2π).

Note that f is continuous on the interval (0, 2π). The

derivative of f is

To determine the critical numbers of f in this interval, set

f'(x) equal to zero.

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20.
Example 2 – Solution cont'd

Because there are no points for which f' does not exist, you

can conclude that x = π/3 and x = 5π/3 are the only critical

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Because there are no points for which f' does not exist, you

can conclude that x = π/3 and x = 5π/3 are the only critical

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21.
Example 2 – Solution cont'd

The table summarizes the testing of the three intervals

determined by these two critical numbers.

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The table summarizes the testing of the three intervals

determined by these two critical numbers.

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22.
Example 2 – Solution cont'd

By applying the First Derivative Test, you can conclude that

f has a relative minimum at the point where

and a relative maximum at the point

as shown in Figure 3.19.

Figure 3.19

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By applying the First Derivative Test, you can conclude that

f has a relative minimum at the point where

and a relative maximum at the point

as shown in Figure 3.19.

Figure 3.19

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