Lecture 8
Increasing and Decreasing
Functions
Graphs
increasing
decreasing
Sometimes increasing, sometimes decreasing
Increasing/decreasing on an
interval
y = f(x)
f is decreasing on [a,b], increasing on [c,d]
f is neither decreasing nor increasing on [a,d], [b,c], [b,d]
Formal Definitions (important)
• The function f is (strictly)
increasing on the
interval (a,b) if for any
a <x<y<b it is true that
f(x)<f(y)
• The function f is (strictly)
decreasing on the
interval (a,b) if for any
a <x<y<b it is true that
f(x)>f(y)
Maybe not strictly
• The function f is
nondecreasing on the
interval (a,b) if for any
a <x<y<b it is true that)
f( x ) f( y )
• The function f is (strictly)
nonincreasing on the
interval (a,b) if for any
a <x<y<b it is true that
f( y ) f( x )
Increasing and Decreasing Don’t
Depend on Derivatives
There are functions that increase on ( ,  ) but cannot be
differentiated anywhere.
Less exotic ones are simply not differentiable in a
lot of places
Relationship Between increasing
(decreasing) and calculus
• f(x) = m x+b is
increasing exactly
when a > 0
• f(x) = m x+b is
decreasing exactly
when a > 0
Idea: If f has a tangent line then f
is increasing when the tangent
line is increasing (i.e. f ‘ (x) > 0)
Theorem: If f ‘ (x) > 0 for every x in the
interval (a,b) then f is increasing on (a,b)
If there are x, y between a and b with x < y and f(x) > f(y) then there is a point
c between x and y where the tangent has negative slope. But then f ‘ (c) < 0
Example: Find the intervals on which
f(x) is increasing (decreasing) if
3
2
f( x ) x  3 x  45 x 5
Solution: Want x such that f ‘ (x) >0 for increasing, f ‘ (x) < 0 for decreasing
2
f ' ( x ) 3 x  6 x 45
Roots = -3, 5
=
3 ( x 3 ) ( x 5 )
Fundamental Fact
• Definition: A critical number of a function f is a
point c at which f ‘(c) = 0 or f ‘ (c) does not exist
• FACT: The sign of the derivative of f does not
change between critical numbers of f.
This gives the following procedure for finding the
intervals of increase and decrease of a function f:
– a. Find all critical numbers.
– Write down the intervals into which they divide the
domain of the function
– Check the sign of the derivative at any point in each
interval – that will be the sign of the derivative at every
point in the interval
Example
3
2
f( x ) x  3 x  45 x 5
2
f ' ( x ) 3 x  6 x 45
=
3 ( x 3 ) ( x 5 )
f ‘ (x) = 0 when x = -3 and c = 5. These are the
critical numbers since f ‘ exists for all x.
The critical numbers divide the domain of f into the intervals
(
,  3
)
(
 3, 5
)
and (
5, 
)
Chose -10 in the first interval, f ‘ (-10) = 3 (-7)(-15) > 0 so f is increasing
on this interval
Chose 0 in the first interval, f ‘ (0) = -45 < 0 so f is decreasing on this
interval
Chose 10 in the first interval, f ‘ (10) = 3(13)(8) > 0 so f is increasing on
this interval
Example:
A particle moves along the x-axis so that its position at time t is given by
x( t ) 3 5 t 4 t
2
What are the times at which the particle is moving left and right?
It is moving to the right when x(t) is increasing and to the left when x(t)
is decreasing.
x ‘ (t) =
 5 8 t
Find critical numbers: x ‘ (t) = 0, t = 5/8
The critical numbers divide the domain into (-infinity, 5/8) and
(5/8, infinity). We choose a convenient member of each.
Take 0 in (-infinity, 5/8), f ‘ (0) = -5 so x ‘ is decreasing on (-infinity, 5/8) so the
particle is moving to the left.
Similarly if we take 2 in (5/8, infinity) then f ‘ (2) >0
So x(t) is increasing on (5/8,infinity) and the particle is moving to the right.