Section 4.4
The Derivative in Graphing and
Applications- “Absolute Maxima
and Minima”
All graphics are attributed to:
• Calculus,10/E by Howard Anton, Irl Bivens, and
Stephen Davis
Copyright © 2009 by John Wiley & Sons, Inc.
All rights reserved.
Introduction
• In this section we will find the highest and
lowest points over the entire “mountain range”
instead of the high and low points in their
immediate vicinity.
• In mathematical terms, we will be looking for
the largest and smallest values of a function over
an interval.
Absolute Extrema
• If a function has an absolute maximum at a
given point in an interval, then the y-value
associated with that point is the largest value of
the function on the interval.
• Likewise for the absolute minimum and the
smallest y-value of the function on the interval.
• There is no guarantee that a function will have
an absolute max. or min. on a given interval.
• See examples on two following slides.
Absolute Extrema Examples
Absolute Extrema Examples con’t
The Extreme Value Theorem
• The extreme value theorem tells us under which
conditions absolute extrema exist.
• We will discuss how to find them on later slides.
• In other words, if the function is continuous on
[a,b], then the absolute extrema occur either at
the endpoints of the interval or at the critical
points inside (a,b).
Graphical Examples
Open Interval Application
• This is also valid on infinite open intervals such
as
.
How to Find the Absolute Extrema
• This is very similar to finding the relative extrema.
• After finding all of the critical points (derivative = 0, solve
for x and non-differentiable points), find out which of them
is the smallest and largest by substitution into f(x).
Polynomial Example
Zero Product
Property
smallest
largest
Absolute Extrema on Infinite Intervals
• You must look at the end behavior of a function when
determining whether or not a function has an absolute
maximum or absolute minimum on −∞, +∞ .
Infinite Interval Example
Absolute Extrema on Open Intervals
• A continuous function may or may not have absolute
extrema on an open interval.
• There are certain conditions that will help determine
whether or not they exist.
Examples
• See examples on pages 270-271
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