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) = x2 - 3x, 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) = x3 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))).