Mean Value Theorems (MVTs) are foundational results in differential calculus that connect function values to derivative values. They are essential for proving inequalities, counting roots, and establishing monotonicity — all frequent JEE topics.
Rolle's Theorem requires three conditions: (1) f continuous on [a,b], (2) f differentiable on (a,b), (3) f(a) = f(b). The conclusion: f'(c) = 0 for some c in (a,b). Geometrically, a smooth curve starting and ending at the same height must have a horizontal tangent somewhere in between. All three conditions are necessary — |x| on [-1,1] satisfies (1) and (3) but fails (2) at x=0, and indeed has no horizontal tangent.
Lagrange's MVT (LMVT) drops the equal-endpoint condition: if f is continuous on [a,b] and differentiable on (a,b), then f'(c) = [f(b)-f(a)]/(b-a) for some c in (a,b). Geometrically, the tangent at c is parallel to the secant joining the endpoints. LMVT generalizes Rolle's — when f(a) = f(b), the ratio becomes 0, recovering Rolle's conclusion.
Cauchy's MVT generalizes further: for f, g continuous on [a,b] and differentiable on (a,b) with g' != 0, we get f''(c) = [f(b)-f(a)]/[g(b)-g(a)]. Setting g(x) = x recovers LMVT. Cauchy's MVT is the theoretical foundation for L'Hopital's Rule.
Key applications of Rolle's include root counting (n roots of f implies at least n-1 roots of f'), uniqueness proofs (f' > 0 everywhere means at most one root), and the auxiliary function technique (to prove f'(c) + kf(c) = 0, apply Rolle's to phi(x) = e^(kx)f(x)).
LMVT applications include proving inequalities (|sin a - sin b| <= |a-b| since |cos c| <= 1), bounding function values (f(1) = 10, |f'| <= 3 implies |f(4)-10| <= 9), and establishing monotonicity (f' > 0 implies f strictly increasing, proved via LMVT).
Darboux's Theorem states that derivatives always have the intermediate value property: f'(c) takes every value between f'(a) and f'(b). This means derivatives cannot have jump discontinuities — any discontinuity must be oscillatory. This constrains which functions can be derivatives.
Taylor's Theorem extends MVT: f(x) = f(a) + f'(a)(x-a) + ... + f^(n)(c)(x-a)^n/n!. The n=1 case is exactly LMVT.