Statement and Conditions: Rolle's theorem requires three conditions: continuity on [a,b], differentiability on (a,b), and f(a) = f(b). The conclusion guarantees f'(c) = 0 for at least one c in (a,b).

Proof sketch: By the Extreme Value Theorem, f attains its maximum M and minimum m on [a,b]. If M = m, f is constant and f'(x) = 0 everywhere. If M > m, at least one extremum occurs at an interior point c (since f(a) = f(b), not both max and min can occur only at endpoints). At this interior extremum, f'(c) = 0 by Fermat's theorem.

Why each condition is necessary:

• Without continuity: f could jump, missing its extremum (e.g., f(x) = 1 for x in (0,1), f(0) = f(1) = 0 — no interior extremum).
• Without differentiability: The extremum could occur at a non-differentiable point (e.g., |x| on [-1,1]).
• Without f(a) = f(b): The function could be monotone (e.g., f(x) = x on [0,1], f' = 1 > 0).

Multiple values of c: Rolle's guarantees at least one c, but there can be many. f(x) = sin(x) on [0, 2pi] gives c = pi/2 and c = 3pi/2.

Root counting application: If f has roots at x1 < x2 < ... < xn, apply Rolle's on [x1,x2], [x2,x3], ..., [x(n-1),xn] to get at least n-1 roots of f'. Repeating: f^(k) has at least n-k roots.

Contrapositive for uniqueness: If f' has at most m zeros, then f has at most m+1 zeros. Strongest form: if f'(x) != 0 for all x, f has at most 1 zero.