Trap 1: Confusing critical point with extremum. A critical point is merely a candidate — it may not be an extremum. Example: f(x) = $x^3$ has f'(0) = 0, but x = 0 is an inflection point, not a max or min. Always verify with a sign change test.

Trap 2: Forgetting endpoints in global extrema. On a closed interval, the global max or min often occurs at an endpoint, not at a critical point. Students who only check critical points miss this. Example: f(x) = $x^3$ - 3x on [0, 3] has a critical point at x = 1 giving f(1) = -2, but the global max is f(3) = 18 at the endpoint.

Trap 3: Second derivative test giving f''(c) = 0. When f''(c) = 0, many students incorrectly conclude "no extremum." In reality, the test is simply inconclusive. The function f(x) = $x^4$ has f'(0) = 0 and f''(0) = 0, but x = 0 is a minimum (verified by f'(x) changing from - to +). Always fall back to the first derivative test.

Trap 4: Ignoring points where f'(x) does not exist. Critical points include not just where f'(c) = 0 but also where f' is undefined. Example: f(x) = |x| has a minimum at x = 0 where f'(0) does not exist. Students who only solve f'(x) = 0 miss such points.

Trap 5: Applying Rolle's/MVT without checking conditions. Rolle's Theorem requires f(a) = f(b). MVT requires continuity on [a, b] and differentiability on (a, b). If f has a cusp or discontinuity in the interval, the theorem does not apply. JEE traps include functions like f(x) = |x - 1| on [0, 2] where f is not differentiable at x = 1.

Trap 6: Sign chart errors with even-multiplicity roots. If f'(x) = (x - 1)^2(x - 3), the sign of f' does NOT change at x = 1 (even multiplicity), so x = 1 is not an extremum. Only x = 3 (odd multiplicity, sign change from - to +) is a minimum.

Trap 7: Incorrect domain in optimization. When setting up an optimization problem, failing to identify the correct domain can lead to extraneous or missed solutions. Example: for an open box cut from a sheet of side a, the cut size x must satisfy 0 < x < a/2, not just x > 0.

Trap 8: Concavity vs. inflection confusion. f''(c) = 0 does not guarantee an inflection point — f'' must change sign. Example: f(x) = $x^4$ has f''(0) = 0 but f'' > 0 on both sides, so x = 0 is not an inflection point.

Trap 9: Treating "increasing" as "non-decreasing." In JEE, "strictly increasing" means f'(x) > 0 (with possible isolated zeros), while "non-decreasing" allows f'(x) = 0 on intervals. Confusing these leads to wrong parameter ranges.