Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test

Section 3.3 – Increasing and Decreasing Functions and the

First Derivative Test

How Derivatives Affect the Shape of the Graph

Many of the applications of calculus depend on our ability to deduce facts about a function f from in information concerning its derivatives. Since the derivative of f represents the slope of tangents lines, it tells us the direction in which the curve proceeds at each point. Thus, it should seem reasonable that the derivative of a function can reveal characteristics of the graph of the function.

Increasing and Decreasing Functions

The function f is strictly increasing on an interval I if f (x1) < f (x2) whenever x1 < x2.

The function f is strictly Decreasing on an interval I if f (x1) > f (x2) whenever x1 < x2.

f(x)

x

D

AC

B IncreasingDecreasing

How the Derivative is connected to Increasing/Decreasing Functions

When the function is increasing, what is the sign (+ or –) of the slopes of the tangent lines?

When the function is decreasing, what is the sign (+ or –) of the slopes of the tangent lines?

f(x)

x

D

AC

B

POSITIVE Slope

NEGATIVE Slope

Test for Increasing and Decreasing Functions

Let f be differentiable on the open interval (a,b)

If f '(x) > 0 on (a,b), then f is strictly increasing on (a,b).

If f '(x) < 0 on (a,b), then f is strictly decreasing on (a,b).

If f '(x) = 0 on (a,b), then f is constant on (a,b).

Procedure for Finding Intervals on which a Function is increasing or Decreasing

If f is a continuous function on an open interval (a,b). To find the open intervals on which f is increasing or decreasing:

1. Find the critical numbers of f in (a,b) AND all values (a,b) of x in that make the derivative undefined.

2. Make a sign chart: The critical numbers and x-values that make the derivative undefined divide the x-axis into intervals. Test the sign (+ or –) of the derivative inside each of these intervals.

3. If f '(x) > 0 in an interval, then f is increasing in that same interval. If f '(x) < 0 in an interval, then f is decreasing in that same interval.

4. State your conclusion(s) with a “because” statement using the sign chart.A sign chart does NOT stand on its own.

Example 1Use the graph of f '(x) below to determine when f is

increasing and decreasing.

(-∞,-1)Increasing: Decreasing: (-1,3)U (3,∞)

f is increasing when the

derivative is positive.

f ' (x)

x

f is decreasing when the

derivative is negative.


derivative is positive.

White Board ChallengeThe graph of f is shown below. Sketch a graph of

the derivative of f.

White Board ChallengeFind the critical numbers of:

3 22 9 12f x x x x

2 1x or x

Example 2Find where the function is

increasing and where it is decreasing. 4 3 23 4 12 5f x x x x

4 3 2' 3 4 12 5ddxf x x x x

Find the derivative.

3 2' 12 12 24f x x x x

Find the critical numbers/where the derivative is undefined

3 20 12 12 24x x x Find where the derivative is 0 or undefined

20 12 2x x x

0 12 2 1x x x 0, 2, 1x

-1 20

Find the sign of the derivative on each interval.

2x 0.5x 1x 3x ' 2 96f

' 0.5 7.5f ' 1 24f

' 3 144f

Answer the question

Domain of f:All Reals

'f x

Example 2: Answer

The function is increasing on (-1,0)U(2,∞) because the first

derivative is positive on this interval.

The function is decreasing on (-∞,-1)U(0,2) because the first

derivative is negative on this interval.

Example 3Use the graph of f (x) below to determine when f is

increasing and decreasing.

(-∞,-3)Increasing: Decreasing: (-3,∞)


function’s outputs are getting larger

as the input increases.

f (x)

x

f is decreasing when the

function’s outputs are getting

smaller as the input increases.

Notice how the function changes from increasing to decreasing at x=-3. But since -3 is not

in the domain of the function, it is not a critical point. Thus, critical points are

not the only points to include in sign

charts.

Example 4Find where the function is increasing and where

it is decreasing. 1

1xf x

2

1 1 1 1

1'

d ddx dxx x

xf x


211

'x

f x

Find the critical numbers/where the derivative is undefinedFind where the derivative is 0 or undefined

-1


2x 0x ' 2 1f ' 0 1f

Answer the question

Domain of f : All Reals except -1

The function does not have any critical points: the derivative is never equal to 0 and the

derivative is only undefined at a point not in the functions domain (x=-1).

Even though -1 is not a critical point, it can still be a point where a function changes from increasing to decreasing. ALWAYS include every x value that

makes the derivative undefined on a sign chart (even if its not a critical point).

'f x

Example 4: Answer

The function is decreasing on (-∞,-1)U(-1,∞) because the

first derivative is negative on this interval.

Make sure not to include -1 in the interval because it is not in the domain of the function.

White Board ChallengeFind the maximum and minimum values

attained by the given function on the indicated closed interval:

4 ; 1, 4xf x x

max : 5 @ 1,4min : 4 @ 2

xx

Critical Values and Relative Extrema

Remember that if a function has a relative minimum or a maximum at c, then c must be a critical number of the function. Unfortunately not every critical number results in a relative extrema.

A new calculus method is needed to determine whether relative extrema exist at a critical point and if it is a maximum or minimum.

How the Derivative is connected to Relative Minimum and Maximum

When a critical point is a relative maximum, what are the characteristics of the function?

When a critical point is a relative minimum, what are the characteristics of the function?

f(x)

x

D

AC

B

The function changes from increasing to decreasing

The function changes from decreasing to increasing

The First Derivative TestSuppose that c is a critical number of a continuous function

f(x).

(a) If f '(x) changes from positive to negative at c, then f(x) has a relative maximum at c.

f(x)

xc

f '(x) < 0 f '(x) > 0

Relative Maximum

The First Derivative TestSuppose that c is a critical number of a continuous function

f(x).

(b) If f '(x) changes from negative to positive at c, then f(x) has a relative minimum at c.

f(x)

xc

f '(x) > 0 f '(x) < 0

Relative Minimum

The First Derivative TestSuppose that c is a critical number of a continuous function f(x).

(c) If f '(x) does not change sign at c (that is f '(x) is positive on both sides of c or negative on both sides), then f(x) has no relative maximum or minimum at c.

f(x)

x

f '(x) < 0 f '(x) > 0 No Relative

Maximum or

Minimumf '(x) > 0

f(x)

xc c

f '(x) < 0

Example 1Use the graph of f '(x) below to determine where f has a

relative minimum or maximum.

@ -1Relative Maximum: Relative Minimum: @ 3

Find the Critical Numbersf ' (x)

x

-1 32x 0x 4x

Make a sign chart and Find the sign of the derivative on each

interval.

Apply the First Derivative Test. 'f x

Example 2Find where the function on 0≤x≤2π has

relative extrema. 2sinf x x x

' 2sinddxf x x x


' 1 2cosf x x

Find the critical numbers

0 1 2cos x Find where the derivative is 0 or undefined

1 2cos x 12 cos x

2 43 3,x

2π/3 4π/3


1x 3x 5x ' 1 2.08f

' 3 0.98f ' 5 1.57f

Answer the question

Domain of f:0≤x≤2π

0 2π 'f x 2 2

3 33f Find the value of the function:

4 43 3 3f

Example 2: Answer

The function has a relative maximum of 3.826 at x = 2π/3 because the first derivative changes from positive to

negative values at this point.

The function has a relative minimum 2.457 at x = 4π/3 because the first

derivative changes from negative to positive values at this point.

Example 3Find the relative extrema values of . 2 31 3 3f x x x

2 31 3' 3ddxf x x x


2 3 1 32 3 1 31 23 3' 3 3f x x x x x

Find the critical numbers

2 3 1 3

2 3 1 3

3 23 3 3

0 x xx x

Find where the derivative is 0 or undefined

2 3 1 3

2 3 1 3

3 23 3 3

x xx x

3 3 6x x

1x


2 3 1 3

2 3 1 3

3 23 3 3

' x xx x

f x

3 9 6x x 9 9x

The derivative

is undefined at x=-3,0

-3 0-14x 2x 0.5x 1x ' 4 1.19f

' 2 0.63f ' 0.5 0.58f

' 1 1.26f

NOTE: 0 is not a relative extrema since the derivative

does not change sign.

Domain of f:All Reals

Answer the question

'f x

Example 3: Answer

The function has a relative maximum of 0 at x = -3 because the first derivative changes

from positive to negative values at this point.

The function has a relative minimum of -1.587 at x = -1 because the first

derivative changes from negative to positive values at this point.

3 0f First find the value of the function:

2 31 2f Now answer the question:

White Board ChallengeFind the intervals on which the function below

is increasing or decreasing.

31f x x x

14

14

increasing :decreasing :

xx

BONUS: How many critical numbers are

there?

Documents

Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test