Statement: If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) with f'(c) = [f(b) - f(a)]/(b - a).

Geometric Meaning: There is a point where the tangent is parallel to the secant joining (a,f(a)) and (b,f(b)).

Applications in JEE:

1. Proving inequalities: If f'(x) >= m on [a,b], then f(b) - f(a) >= m(b-a).
2. Bounding function values: |sin x - sin y| <= |x - y| (since |cos c| <= 1).
3. Establishing bounds on integrals and derivatives.

Example: Prove |sin(a) - sin(b)| <= |a - b|. By LMVT: sin(a) - sin(b) = cos(c)(a - b) for some c. |cos(c)| <= 1. Done.