Section 13.1 – 13.2
Increasing/Decreasing Functions
and Relative Extrema
Facts


If f ’(x) > 0 on an interval (a,b), then f (x) is
increasing on (a,b).
If f ’(x) < 0 on an interval (a,b), then f (x) is
decreasing on (a,b).
Definition:

A number c for which f ’(c) = 0 or f ’(c) = undefined
is called the critical number (critical value).
Example:
Find the intervals where the function
is increasing/decreasing
2
3
f ( x)  5  3 x  x
Definition
 A function f has a relative maximum (or local max)
at c if f (c) > f (x) for all x near c.
 A function f has a relative minimum (or local min)
at c if f (c) < f (x) for all x near c.
The First Derivative Test:
If f ’(c) changes from + to – at c, then f has a local
maximum at c.
If f ’(c) changes from – to + at c, then f has a local
minimum at c.
 No sign change at c means no local extremum
(maximum or minimum)
How to find local max/min and
interval of increasing/decreasing:
1)
2)
3)
Find all critical values by solving f ’(x) = 0 or
f ’(x) = undefined
Put all critical values on the number line and
use test values to determine the sign of the
derivative for each interval.
Determine the interval of increasing/decreasing
based on the sign of derivative.
Examples
Find the intervals of increase/decrease
and all local extrema.
f ( x)  4 x  9 x  30 x  6
3
2
x
g ( x)  2
x 3
2
h( x)  x e
2 x
y  6x 3  4x 1
2
Examples
A small company manufactures and sells bicycles.
The production manager has determined that the
cost and demand functions for q (q > 0) bicycles per
week are
1 3
C (q )  10  5q  q
60
p  D(q)  90  q
where p is the price per bicycle.
a) Find the (weekly) revenue function.
b) Find the maximum weekly revenue.
c) Find the maximum weekly profit.
d) Find the price the company should charge to
realize maximum profit.
7