Rolle's Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) such that f'(c) = 0.

Geometric meaning: If the function starts and ends at the same height, somewhere in between the tangent is horizontal.

Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c) = [f(b)-f(a)]/(b-a).

Geometric meaning: Somewhere between a and b, the tangent line is parallel to the secant line joining (a,f(a)) and (b,f(b)).

Key applications of MVT in JEE:

1. Proving inequalities: Show sin(x) < x for x > 0 using MVT on sin on [0,x].
2. Finding c: Given f and interval, solve f'(c) = [f(b)-f(a)]/(b-a).
3. Proving existence: Show that f'(c) = k has a solution in some interval.

Rolle's Theorem application: To prove an equation g(x) = 0 has a root in (a,b), try to find f such that f' = g and f(a) = f(b). Then by Rolle's, f'(c) = g(c) = 0.