A necessary condition for an interior optimum

From your previous study of mathematics, you probably know that if the function f is differentiable there is a relationship between the solutions of the problem

max_x f (x) subject to x Î [a,b].

and the points at which the first derivative of f is zero---the stationary points of f . What precisely is this relationship?

Consider the cases in the three figures.

In the left figure, the unique stationary point x* is the global maximizer.
In the middle figure, there are three stationary points: x*, x¢, and x¢¢. The point x* is the global maximizer, while x¢ is a local (though not global) minimizer and x¢¢ is a local (but not global) maximizer.
In the right figure, there are two stationary points: x¢ and x¢¢. The point x¢ in neither a local maximizer nor a local minimizer; x¢¢ is a global minimizer.

We see that

a stationary point is not necessarily a global maximizer, or even a local maximizer, or even a local optimizer of any sort (maximizer or minimizer) (consider x¢ in the right-hand figure)
a global maximizer is not necessarily a stationary point (consider a in the right-hand figure).

That is, being a stationary point is neither a necessary condition nor a sufficient condition for solving the problem. So what is the relation between stationary points and maximizers?

Although a maximizer may not be a stationary point, the only case in which it is not is when it is one of the endpoints of the interval [a,b] on which f is defined. That is, any point interior to this interval that is a maximum must be a stationary point.

Proposition
Let f be a differentiable function of a single variable defined on the interval [a, b]. If x Î (a, b) is a local or global maximizer or minimizer of f then f ¢(x) = 0.

This result gives a necessary condition for x to be a maximizer (or a minimizer) of f : if it is a maximizer (or a minimizer) and is between a and b then x is a stationary point of f . The condition is obvously not sufficient for a point to be a maximizer---the condition is satisfied also, for example, at points that are minimizers. Since the first-derivative is involved, we refer to it as a first-order condition.

Thus among all the points in the interval [a,b], only a, b, and the stationary points of f can possibly be maximizers of f . Since most functions have a relatively small number of stationary points, a reasonable way of finding maximizers is the following:

find all the stationary points of f (the points x for which f ¢(x) = 0) and calculate the values of f at each of these points
find the values of f at the endpoints a and b of its domain
the largest of all these values is the maximum
the smallest of all these values is the minimum

Exercise