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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.
(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
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 viceversa.
Slide 2.1- 7
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Find the critical numbers for
f ( x)  x  x  5 x  4
3
2
f ( x)  6 x
2/3
Steps For Finding Increasing and
Decreasing Intervals of a Function
1) Find the derivative
2) 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.
Example 1:
Find the increasing and
decreasing intervals for the
funtion given by
f ( x)  2 x  3x  12 x  12.
3
Slide 2.1- 10
2
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Example 1 (continued): Find Derivative
And set it = 0
6x 2  6x  12  0
x2  x  2
(x  2)(x  1)
x2
or
 0
 0
x  1
These two critical values partition the number line into
3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).
A
B
-1
Slide 2.1- 11
C
2
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Example 1 (continued):
3rd analyze the sign of f (x) in each interval.
Interval
A
C
B
-1
x
2
Test Value
x = –2
x=0
x=4
Sign of
f (x)
+
–
+
Result
Slide 2.1- 12
f is increasing f is decreasing on
on (–∞, –1]
[–1, 2]
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
f is increasing
on [2, ∞)
Find the intervals where f is increasing
and decreasing.
f ( x)  x  5 x  6
2
Since f ’(x) = 2x+5 it follows that
f is increasing when 2x+5>0 or
when x>-2.5 which is the interval
(2.5, )
It is decreasing on (,2.5)
It is increasing on (2.5, )
Find the intervals where the function
is increasing and decreasing
f ( x)  x
3
Determine the critical numbers for each
function and give the intervals where the
function is increasing or decreasing.
f ( x)  ( x  3)
x3
f ( x) 
x4
2/3
x2
f ( x)  2
x 5
1 2
f ( x)  x  x  x  19
2
3
A product has a profit function of
P( x)  0.01x  60 x  500
2
for the production and sale of x
units. Is the profit increasing or
decreasing when 100 units have
been sold?
Since C(x) is the cost for producing x
units, the average cost for producing
x units is C(x) divided by x.
C ( x)
C ( x) 
x
The marginal average cost would be
found by taking the derivative.
Suppose a product has a cost
function given by
C ( x)  500  54 x  .03x
where 0  x  1000
2
Find the average cost function.
Over what interval is the average
cost decreasing?
Suppose the total cost C(x), in dollars, to manufacture a
quantity x of weed killer, in hundreds of liters, is given by
C ( x)  x3  2 x 2  8x  50
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. 0  x  3300
Determine the interval on which the profit function is
increasing.
C( x)  0.32 x 2  0.00004 x3
R( x)  0.848 x 2  0.0002 x3
(0, 2200)