Example 4

, a = -1, b = 1. Then f(-1) = f(1) = 0. The function f is continuous on [-1, 1] and has a derivative at each point of (-1,1), as Rolle's Theorem requires (Figure 3.8.14). Note, however, that f'(x) does not exist at either endpoint, x = -1 or x = 1. By Rolle's Theorem there is a point c in (-1, 1) such that f'(c) = 0, c = 0 is such a point, because

, f'(0) = 0.

Figure 3.8.14