Section 1: Monotonicity and Intervals of Increase/Decrease

Core technique: Compute f'(x), factor it completely, find zeros, then construct a sign chart. The sign of f' in each interval determines whether f is increasing (+) or decreasing (-). For functions with parameters (e.g., "find values of a for which f is increasing everywhere"), set up the condition f'(x) >= 0 for all x. For quadratic f'(x), this means discriminant <= 0 with positive leading coefficient. For functions involving |x| or piecewise definitions, analyze each piece separately and check continuity at junction points.

Section 2: Local Extrema — First and Second Derivative Tests

Step-by-step approach: (1) Find f'(x) and set equal to zero to get critical points. (2) Also check where f'(x) is undefined. (3) Apply the first derivative test by checking sign of f' on either side, OR apply the second derivative test by computing f''(c). The first derivative test is universally reliable. The second derivative test fails when f''(c) = 0, requiring fallback to the first derivative test or the higher-order derivative test. JEE frequently tests functions like f(x) = $x^n$ * e^(-x) or f(x) = $x^2$ * ln(x) where careful algebra is needed.

Section 3: Global Extrema on Closed Intervals

The Extreme Value Theorem guarantees existence. Method: list all critical points in the open interval, evaluate f at each critical point and both endpoints, then compare. Common JEE twist: the function is defined piecewise on the interval, so you must find critical points in each piece and also check the junction points (which may be critical points where f' is undefined).

Section 4: Rolle's Theorem and Mean Value Theorem

For Rolle's: verify the three conditions (continuous on [a, b], differentiable on (a, b), f(a) = f(b)), then solve f'(c) = 0 to find c. For MVT: verify two conditions, compute the slope [f(b) - f(a)]/(b - a), then solve f'(c) = slope. JEE application: use MVT to prove inequalities. Example: prove $\frac{b - a}{sqrt}$(1 - $a^2$) < arcsin(b) - arcsin(a) < $\frac{b - a}{sqrt}$(1 - $b^2$) for 0 < a < b < 1. Apply MVT to f(x) = arcsin(x) on [a, b].

Section 5: Concavity and Points of Inflection

Find f''(x), determine where it equals zero or is undefined, then check sign change across those points. Only points where f'' actually changes sign are inflection points. For curve sketching: combine information from f' $\frac{increase}{decrease}$ and f'' (concavity) to determine the shape in each interval.

Section 6: Tangent, Normal, and Optimization

For tangent/normal problems: compute f'(x0) at the given point, write the line equation, then use any additional condition (parallel to a line, passes through a point, etc.) to find unknowns. For optimization: express the objective in terms of one variable, differentiate, find critical points, and verify. Always check that the critical point is within the domain and is actually a maximum/minimum (not just a critical point).