文档介绍：Section
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Finding Maximum and
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Problems involving finding the maximum or minimum value of a quantity occur frequently
in mathematics and in the applications of mathematics. pany may want to maximize
its profit or minimize its costs; a farmer may want to maximize the yield from his crop
or minimize the amount of irrigation equipment needed to water his fields; an airline may
want to maximize its fuel efficiency or minimize the length of its routes. Methods for
solving some optimization problems are putationally intense that they challenge,
and sometimes even go beyond, the puters currently available. An example of
such a problem is the famous traveling salesman problem, in which a salesman wishes to
visit a certain set of cities using the shortest possible route. In this section we will not
consider problems of this type, but rather we will confine ourselves to problems involving
continuous functions of a single independent variable.
Closed intervals
We will start with the simplest case. Suppose f is a continuous function on a closed interval
[a, b]. From the Extreme Value Theorem we know that f attains both a maximum value
and a minimum value on the interval. We now look for candidates at which these values
might occur. To start, an extreme value could occur at one of the endpoints. For example,
the maximum value of f(x) = x2 on [0, 1] occurs at x = 1. If an extreme value occurs in
the open interval (a, b) at a point c where f is differentiable, then f has a local extremum
at c and so, from our work in Section , we know that f 0(c) = 0. For example, the
minimum value of f(x) = x2 on [−1, 1] occurs at x = 0 and f 0(0) = 0. Finally, the only
other candidates for the locations of extreme values would be points where f 0 is undefined.
For example, the minimum value of f(x) = |x| on [−1, 1] occurs at x = 0, where f 0 is not
defined. Hence we are led to t