Critical Points

Definition A function f(x) is said to have a local maximum at c iff there exists an interval I around c such that Analogously, f(x) is said to have a local minimum at c iff there exists an interval I around c such that A local extremum is a local maximum or a local minimum. Using the definition of the derivative, we can easily show that:

Example. Consider the function f(x) = x3. Then f'(0) = 0 but 0 is not a local extremum. Indeed, if x 0, then f(x) > f(0). Therefore the conditions do not imply in general that c is a local extremum. So a local extremum must occur at a critical point, but the converse may not be true.