Monotonicity and Inverse Functions: If f is strictly monotonic on I, then f has an inverse f^(-1) on f(I). Strict monotonicity is sufficient for invertibility. This is why sin^(-1) exists: sin is strictly increasing on [-pi/2, pi/2].

Monotonicity and Number of Roots: If f is strictly increasing on (a,b) and f(a) < 0 < f(b), then f has exactly one root in (a,b). Use this to count roots of equations.

Monotonicity and Inequalities: If f'(x) > 0 on (a, inf) and f(a) = 0, then f(x) > 0 for all x > a. This is the standard technique for proving inequalities.

MVT and Definite Integrals: The MVT for integrals states: integral from a to b of f(x)dx = f(c)*(b-a) for some c in (a,b). This connects derivatives (MVT) to integrals.

Extrema and Linear Algebra: Quadratic optimization (max/min of $ax^{2+bx+c}$) connects to vertex form: vertex at x = -$\frac{b}{2a}$. Maximum if a < 0, minimum if a > 0.

Cross-topic JEE problems often combine monotonicity with:

• Limits (finding ranges of convergence)
• Integration (estimating integrals using bounds)
• Differential equations (stability analysis)