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).

Physical interpretation: If a car travels from position f(a) at time a to position f(b) at time b, its average velocity is [f(b)-f(a)]/(b-a). LMVT says the instantaneous velocity must equal this average at some moment c.

Relation to Rolle's: LMVT is Rolle's applied to g(x) = f(x) - (f(b)-f(a))/(b-a). Then g(a) = f(a) = g(b), so g'(c) = 0 gives f'(c) = [f(b)-f(a)]/(b-a).

Key consequence: If f'(x) = 0 for all x in (a,b), then f is constant on [a,b]. Proof: For any two points x1, x2, f(x2) - f(x1) = f'(c)(x2-x1) = 0.