Critical Point Theorem

Note that on the graph presented at the start of this section, f had local extrema at x = b, x = c, and x = d.

Figure %: Graph of function f on the interval [a, e]

It seems as though the tangent to the graph at each of these points is horizontal. It is in fact always the case that: if f has a local extrema at b and f'(b) exists, then f'(b) = 0.

Sometimes, it is also possible for a continuous function to have a local extremum at a point where the derivative does not exist. For example, the function f (x) =|x - b| has a local min at x = b.

Figure %: f (x) =|x - b|

Note that the derivative, f'(b), does not exist in this case.

We can combine these two observations into a single theorem called the Critical Point Theorem. A critical point of a function f occurs where f'(x) = 0 or f'(x) is undefined. Then the statement of the critical point theorem is that if f has a local extremum at x = b, then (b, f (b)) is a critical point.

Note that the converse of this theorem is not true, i,e, it is not the case that all critical points are local extrema. For example, in the graph below, the point x = b has a horizontal tangent, so f'(b) = 0, but f does not have a local extremum at b:

Figure %: The converse of the critical point theorem is not necessarily true