Method 1: Monotonicity To prove f(x) > g(x) for x > a: Define h(x) = f(x) - g(x). Show h'(x) > 0 for x > a and h(a) >= 0. Then h(x) > 0 for all x > a.

Example: Prove sin x < x for x > 0. h(x) = x - sin x. h'(x) = 1 - cos x >= 0. h(0) = 0. So h(x) >= 0 for x >= 0, with equality only at x = 0. Hence sin x < x for x > 0.

Method 2: MVT To show |f(b) - f(a)| <= M|b-a|: By MVT, f(b)-f(a) = f'(c)(b-a). If |f'(c)| <= M, done.

Example: Prove |sin a - sin b| <= |a - b|. By MVT: sin a - sin b = cos(c)(a-b). |cos c| <= 1. So |sin a - sin b| <= |a-b|.

Method 3: Second derivative for convexity/concavity If f''(x) > 0 (convex), then f(x) >= f(a) + f'(a)(x-a) for all x (tangent line lies below curve).

JEE Pattern: These inequality proofs typically require defining an appropriate auxiliary function and showing it's monotonic.