Optimization is nothing more than finding the minimum or maximum values of a function within
a specified part of its domain. For instance, a function
*f* (*x*)
may represent a quantity of
practical significance (profit, revenue, temperature, efficiency) with the variable
*x*
representing a quantity that can be controlled (expenditures, investment, throttle, length of
work day). Then an approximate formula for
*f* (*x*)
, for instance
*f* (*x*) = *x*
^{2} - 3*x*
, might
make sense for values of
*x*
that have no real significance (such as negative length), so
the domain of
*f*
must be artificially restricted to fit with the practical application.

To find the global maximum or minimum of
*f*
, if it exists, one must check determine the
positions of the local maxima and local minima, and compare these to the values of
*f*
at the endpoints of its domain, if there are any.

It may happen that a function, such as
*f* (*x*) = *x*
^{3}
with domain
[3, 4]
, does not have any
critical points, but attains a global maximum at an endpoint -- in this case
*f* (4) = 64
. It
may also happen that a function has critical points but does not have a global maximum or
minimum, for instance
*f* (*x*) =
with domain
(- 1, 1)
. The latter phenomenon
uses the "openness" of the domain
(- 1, 1)
in an essential way; the function has no maximum
or minimum exactly because it approaches
±∞
at the omitted endpoints
±1
.

The most convenient setting for optimization problems is then a differentiable function
*f*
whose domain is a *closed* interval
[*a*, *b*]
. In this case,
*f*
has both a global
maximum and a global minimum, each of which is either a critical point or a boundary point
(i.e.
(*a*, *f* (*a*))
and
(*b*, *f* (*b*))
).

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