5.3 Increasing and Decreasing Functions
First Derivative Test
­ will learn how the derivative may be used to classify relative extrema
as either relative min. or relative max.
review:
as x increases and f(x) increases, f(x) is INCREASING
as x increases and f(x) decreases, f(x) is DECREASING
1
Theorem 5.5 Test for INC. or DEC. functions
Let f be continuous on [a, b] and differential on (a, b).
1. If f '(x) > 0, then f is INC. on the interval (a, b).
2. If f '(x) < 0, then f is DEC. on the interval (a, b).
3. If f '(x) = 0, then f is CONSTANT on the interval (a, b).
Dec.:________________
Inc.:_________________
Constant:_____________
ex.
State the open intervals on which f(x) = x3 ­ (3/2)x2 is inc./dec.
* note that f is continuous and differentiable for all x
HW
pg. 335
#2­10 (E)
2
­ A function is STRICTLY MONOTONIC on an interval if it is
either increasing on the entire interval or decreasing on the entire
interval.
ex.
f(x) = x3 is STRICTLY MONOTONIC (inc.)
ex.
NOT Strictly Monotonic
3
The First Derivative Test
Let c be a critical number of f ' (x):
1. If f '(x) changes from neg. to pos. at c ­­> rel. min. at (c, f(c))
2. If f '(x) changes from pos. to neg. at c ­­> rel. max. at (c, f(c))
3. If f '(x) is pos. or neg. on both sides of c ­­> plateau pt. at (c, f(c))
1. If f '(x) changes from neg. to pos. at c ­­> rel. min. at (c, f(c))
2. If f '(x) changes from pos. to neg. at c ­­> rel. max. at (c, f(c))
3. If f '(x) is pos. or neg. on both sides of c ­­> plateau pt. at (c, f(c))
* The only place that a continuous function may
change from INC. to DEC. or DEC. to INC.
is at a critical #.
4
ex.
Find the relative extrema for:
5
SUMMARY:
1) Relative MIN. occurs where the derivative
changes from neg. to pos.
Relative MIN occurs where the function
changes from dec. to inc.
2) Relative MAX. occurs where the derivative
changes from pos. to neg.
Relative MAX occurs where the function
changes from inc. to dec.
6
* remember, a graphing calculator by itself may give misleading information about the
relative extrema of a graph, however, used
in conjunction with an analytic approach, it
is a helpful tool to reinforce your conclusions
ex.
Use your graphing calculator to find the
relative maximum for:
(set the window at ­5, 5, ­5, 5)
­ it appears the relative max. has a value of 1
and occurs at x = 1
* now set the window at:
xmin=0.9, xmax=1.1, xscl=0.1
ymin=­0.5, ymax=1.5, yscl=0.1
* f (1) = 0 and not 1 !!!
7
ex.
Find the relative extrema for:
8
HW
pg. 335
#12, 20, 24, 28, 32
9
10