Pattern 1: LMVT Verification (Moderate Frequency) Given f on [a,b], compute [f(b)-f(a)]/(b-a), solve f'(c) = result, verify c in (a,b). Typically with polynomials or simple functions.

Pattern 2: Inequality via LMVT (Moderate) Prove bounds like |sin a - sin b| <= |a-b| or $e^x$ > 1+x. Apply LMVT and bound the derivative.

Pattern 3: Root Counting (High Frequency) Determine the exact number of real roots. Use f' analysis (critical points), sign of f at critical values, and IVT. Most common JEE MVT application.

Pattern 4: Rolle's Application (Low-Moderate) Given f(a) = f(b), find c. Or use Rolle's to prove existence of solutions to derivative equations.

Pattern 5: Conceptual True/False (Low) "f differentiable, f'(a) > 0, f'(b) < 0 implies f'(c) = 0" — TRUE by Darboux. "f'(c) = 0 implies f has a maximum or minimum at c" — FALSE.

Pattern 6: Auxiliary Function (Rare but High-Value) Construct phi(x) to apply Rolle's. Usually in proof-based or assertion-reasoning questions.