Theorem: If f'(x) > 0 for all x in (a,b), then f is strictly increasing on [a,b].

Proof using LMVT: For x1 < x2 in [a,b], LMVT gives f(x2) - f(x1) = f'(c)(x2 - x1) for some c in (x1, x2). Since f'(c) > 0 and x2 - x1 > 0, we get f(x2) > f(x1).

Theorem: If f'(x) = 0 for all x in (a,b), then f is constant on [a,b].

Theorem: If f'(x) >= 0 on (a,b) and f' is not identically zero on any subinterval, then f is strictly increasing.

Application: To prove f(x) >= g(x) on [a, infinity), define h = f - g, show h(a) >= 0 and h'(x) >= 0. By LMVT, h is non-decreasing, so h(x) >= h(a) >= 0.