at both of the endpoints: f(1) = 1 − = 0 and f(−1) = 1 − = 0. Next, we take the
derivative to find f′(c): f′(x) = , so f′(c) = . This has no solution.
Therefore, remember that it’s very important to check that the function is continuous and
differentiable everywhere on the given interval (it does not have to be differentiable at the
endpoints) when doing Rolle’s Theorem problems. If it is not, then the theorem does not apply.
- c =
Rolle’s Theorem says that if f(x) is continuous on the interval [a, b] and is differentiable
everywhere on the interval (a, b), and if f(a) = f(b) = 0, then there exists at least one number c
on the interval (a, b) such that f′(c) = 0. Here the function is f(x) = x − x and the interval is [0,
1]. First, we check if the function is equal to zero at both of the endpoints: f(0) = (0) − (0) = 0
and f(1) = (1) − (1) = 0. Next, we take the derivative to find f′(c) f(x) =
, so f(c) = . Now, we can solve for c:
= 0 and c = . Note that is in the interval (0, 1), just as we expected.
