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*))).