Statement: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c in (a,b) such that f'(c) = 0.

Why each condition matters:

1. Continuity on [a,b]: Ensures f attains max and min on [a,b] (Extreme Value Theorem). Without this, f could jump over its extremum.
2. Differentiability on (a,b): Ensures tangent line exists at interior points. |x| on [-1,1] satisfies f(-1)=f(1) but f'(0) DNE — no horizontal tangent guaranteed.
3. f(a) = f(b): Without this, we get LMVT instead. The horizontal tangent conclusion requires equal endpoints.

Geometric intuition: A smooth curve that starts and ends at the same height must have at least one peak or valley in between — at that point, the tangent is horizontal.