Page 114 - Calculus Workbook For Dummies

98 Part III: Differentiation

Finding Mount Everest: Absolute Extrema

The basic idea in this section is quite simple. Instead of finding all local extrema like in
the previous sections (all the peaks and all the valleys), you just want to determine the
single highest point and single lowest point along a continuous function in some closed
interval. These absolute extrema can occur at a peak or valley or at an edge(s) of the
interval. (Note: You could have, say, two peaks at the same height so there’d be a tie
for the absolute max; but there would still be exactly one y-value that’s the absolute
maximum value on the interval.)
Before you practice with some problems, look at Figure 7-1 to see two standard
absolute extrema problems (continuous functions on a closed interval) and at Figure 7-2
for four strange functions that don’t have the standard single absolute max and single
absolute min.

1. 2.
y y
abs
max
abs
Figure 7-1: max
Two
abs
standard min abs
min
absolute
x x
extrema 1 2 3 4 1 2 3 4
functions.
open interval closed interval

y no abs max y no abs max

abs abs
min min
x x
1 2 3 4 1 2 3 4
asymptote
open interval closed interval

no abs min no abs max or min
y y
abs
max
Figure 7-2:
Four non-
standard
absolute hole
extrema x x
1 2 3 4 1 2 3 4
functions.
closed interval open interval