Let f(x) = sin(x). This function is differentiable everywhere. Prove that there is some c in (0, 2π) with f'(c) = 0. By looking at the graph of f, determine how many such values of c there are in (0,2π).
We need to check that f satisfies all the hypotheses of Rolle's Theorem.
f(x) = sin(x) is continuous everywhere, so f is continuous on [0,2π].
f is differentiable everywhere, therefore it's differentiable on (0,2π).
f(0) = 0 and f(2π) = 0, so f(0) = f(2π).
Since f satisfies all the hypotheses of Rolle's Theorem, there is some value of c in (0,2π) with f'(c) = 0.By looking at the graph of f(x) = sin(x), we see that there are two such values of c:
If any one of the hypotheses of Rolle's Theorem fails, the conclusion can fail too.