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

Application 1: Proving Inequalities If m <= f'(x) <= M on [a,b], then m(b-a) <= f(b)-f(a) <= M(b-a).

Classic examples:

• |sin a - sin b| <= |a-b|: Use f = sin x, |f'| = |cos| <= 1.
• $\frac{x}{1+x}$ < ln(1+x) < x for x > 0: Apply LMVT to ln(1+t) on [0,x].
• $\frac{b-a}{b}$ < ln$\frac{b}{a}$ < $\frac{b-a}{a}$ for 0 < a < b: Apply to ln(t) on [a,b].

Application 2: Approximation sqrt(26) - 5 = $\frac{1}{2sqrt(c}$) for c in (25,26). Since c > 25: result < 0.1. e^(0.01) - 1 = $e^c$ * 0.01 for c in (0, 0.01). Since c > 0: $e^c$ > 1, so e^(0.01) > 1.01.

Application 3: Monotonicity Proofs f'(x) > 0 on I => f strictly increasing on I. Proof: for x1 < x2, f(x2)-f(x1) = f'(c)(x2-x1) > 0.

Application 4: Lipschitz Condition |f'(x)| <= L implies |f(a)-f(b)| <= L|a-b|. This gives a quantitative bound on how fast f can change.

Application 5: Bounding function values If f(a) and a bound on f' are known, LMVT bounds f(b): f(a) + m(b-a) <= f(b) <= f(a) + M(b-a).