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Chapter 4
Applied Maximum and
Minimum
Optimization problems are one of the most important applica-
tions of differential calculus. Whether we are concerned with
how to get from A to B in the least amount of time, or we wish
to construct a box of maximum volume for a given amount of
material, we are looking for the “best” way to perform a given
task.
The problem of optimization generally reduces to the
problem of finding the maximum or minimum value of a func-
tion subject to a given set of conditions or constraints. In this
chapter we will discuss how to set up a maximum/minimum
problem and solve it to find the optimal solution.
We begin by reviewing a few basic definitions and
theorems.
A function f has an absolute maximum on an interval I
if there exists a number c in I such that f (x) ≤ f (c) for all
x in I.
A similar definition (with the inequality reversed) applies to
an absolute minimum. Note that that absolute maximum or
minimum value is f (c). Its location is x = c.
The existence of an absolute maximum and minimum
under certain conditions is guaranteed by the Extreme Value
Theorem:
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