5.1 Increasing\decreasing Functions

Find critical values of a function

Find increasing/decreasing intervals of a function

A function is increasing when its graph rises as it goes from left to right.

A function is decreasing when its graph falls as it goes from left to right.

inc in

c

dec

(Scary Math) DEFINITIONS:

A function f is increasing over I if, for every a and b in I, if a < b, then f (a) < f (b).

The graph rises from left to right.

A function f is decreasing over I if, for every a and b in I, if a < b, then f (a) > f (b).

The graph falls from left to right

Whether a function is increasing or decreasing is related to the the slope of the tangent line.

The slope of tan line positive - function increasing.

The slope of tan line negative - function is decreasing.

On an interval on which f is defined:

If f(x) > 0 (if the derivative is positive) for all x in an interval I, then f (the function) is increasing over I.

If f(x) < 0 (if the derivative is negative) for all x in an interval I, then f (the function) is decreasing over I.

Find the intervals where f is increasing and decreasing

Slide 2.1- 7

Critical Value or Critical Number

A critical value (or critical number) of a function f is any number c in the domain of f for which the tangent line at (c, f (c)) is horizontal or for which the derivative does not exist.

That is, c is a critical value if f (c) exists and f (c) = 0 or f (c) does not exist.

These are the x values where the function could change from increasing to decreasing or vice-versa.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find the critical numbers for

45)( 23 xxxxf

3/26)( xxf

Steps For Finding Increasing and Decreasing Intervals of a Function

1) Find the derivative2) Find numbers that make the derivative equal to 0,

and find numbers that make it undefined. These are the critical numbers.

3) Put the critical numbers and any x values where f is undefined on a number line, dividing the number line into sections.

4) Choose a number in each interval to test in the first derivative. Make a note of the sign you get.

5) Intervals that have first derivatives that are positive, are increasing, and intervals that have first derivatives that are negative, are decreasing.

Slide 2.1- 10Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

Example 1: Find the increasing and decreasing intervals for the funtion given by

3 2( ) 2 3 12 12.f x x x x

Slide 2.1- 11Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

Example 1 (continued): Find Derivative And set it = 0

These two critical values partition the number line into 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

CB A

2-1

6x2 6x 12 0

x2 x 2 0

(x 2)(x 1) 0

x 2 or x 1

Slide 2.1- 12Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

Example 1 (continued):3rd analyze the sign of f (x) in each interval.

Test Value x = –2 x = 0 x = 4

Sign off (x)

+ – +

Resultf is increasing on (–∞, –1]

f is decreasing on [–1, 2]

f is increasing on [2, ∞)

xInterval

CB A

2-1

Find the intervals where f is increasing and decreasing.

65)( 2 xxxfSince f ’(x) = 2x+5 it follows thatf is increasing when 2x+5>0 orwhen x>-2.5 which is the interval

),5.2(

It is decreasing on It is increasing on

)5.2,( ( 2.5, )

Find the intervals where the function is increasing and decreasing

3)( xxf

Determine the critical numbers for each function and give the intervals where the

function is increasing or decreasing.3/2)3()( xxf

3( )

4

xf x

x

192

1)( 23 xxxxf

5

2)(

2

x

xxf

A product has a profit function of

for the production and sale of x units. Is the profit increasing or decreasing when 100 units have

been sold?

2( ) 0.01 60 500P x x x

Since C(x) is the cost for producing x units, the average cost for

producing x units is C(x) divided by x.

x

xCxC

)()(

The marginal average cost would be found by taking the derivative.

Suppose a product has a cost function given by

Find the average cost function.Over what interval is the average

cost decreasing?

2( ) 500 54 .03

0 1000

C x x x

where x

Suppose the total cost C(x), in dollars, to manufacture a quantity x of weed killer, in hundreds of liters, is given by

3 2( ) 2 8 50C x x x x

a) Where is C(x) increasing?b) Where is C(x) decreasing?

a) Nowhere b) (0, infinity)

A manufacturer sells video games with the following cost and revenue functions (in dollars), where x is the number of games sold.

Determine the interval on which the profit function is increasing.

0 3300x

2 3

2 3

C( ) 0.32 0.00004

( ) 0.848 0.0002

x x x

R x x x

(0, 2200)