Statement: If f has continuous derivatives up to order n on [a,x], then: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n-1)(a)(x-a)^$\frac{n-1}{(n-1)}$! + f^(n)(c)(x-a)^n/n! for some c between a and x. The last term is the Lagrange remainder.

n = 0: f(x) = f(c) — just says f takes every value between f(a) and f(x) (not quite IVT but related). n = 1: f(x) = f(a) + f'(c)(x-a) — this IS LMVT (with c in (a,x)).

JEE application: Bounding errors in polynomial approximation. If |f''(x)| <= M on [a,b], then |f(x) - f(a) - f'(a)(x-a)| <= M(x-a)^2/2.