Method: Apply LMVT to get f(b) - f(a) = f'(c)(b-a) for some c in (a,b), then bound f'(c).

Classic inequality 1: |sin a - sin b| <= |a - b| Apply LMVT to sin x: sin b - sin a = cos(c)(b-a). Since |cos c| <= 1: |sin b - sin a| <= |b-a|.

Classic inequality 2: $\frac{x}{1+x}$ < ln(1+x) < x for x > 0 Apply LMVT to ln(1+t) on [0,x]: ln$\frac{1+x}{x}$ = $\frac{1}{1+c}$ for c in (0,x). Since 0 < c < x: $\frac{1}{1+x}$ < $\frac{1}{1+c}$ < 1. Multiply by x: $\frac{x}{1+x}$ < ln(1+x) < x.

Classic inequality 3: $\frac{b-a}{b}$ < ln$\frac{b}{a}$ < $\frac{b-a}{a}$ for 0 < a < b Apply LMVT to ln(t) on [a,b]: ln(b) - ln(a) = $\frac{1}{c}$(b-a). Since a < c < b: 1/b < 1/c < 1/a.

General strategy: If m <= f'(x) <= M on [a,b], then m(b-a) <= f(b)-f(a) <= M(b-a).