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3.3. GLOBAL OPTIMIZATION
3.3 Global Optimization
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are the differences between finding relative extreme values and global extreme
values of a function?
• How is the process of finding the global maximum or minimum of a function over
the function’s entire domain different from determining the global maximum or
minimum on a restricted domain?
• For a function that is guaranteed to have both a global maximum and global
minimum on a closed, bounded interval, what are the possible points at which
these extreme values occur?
Introduction
We have seen that we can use the first derivative of a function to determine where the
function is increasing or decreasing, and the second derivative to know where the function
is concave up or concave down. Each of these approaches provides us with key information
that helps us determine the overall shape and behavior of the graph, as well as whether
the function has a relative minimum or relative maximum at a given critical number.
Remember that the difference between a relative maximum and a global maximum is
that there is a relative maximum of f at x = p if f (p) ≥ f (x) for all x near p, while
there is a global maximum at p if f (p) ≥ f (x) for all x in the domain of f . For instance,
f
relative max
a
relative min
b
global max
c
Figure 3.17: A function f with a global maximum, but no global minimum.