Mistake 1: Forgetting endpoints for global extrema On [1, 3], f(x) = $x^{3-3x}$ has f'(x) = 3$x^{2-3}$ = 0 at x = 1. But x = 1 is an endpoint! Must evaluate f(1) = -2, f(3) = 18. Global max is at x = 3, not at a critical point.

Mistake 2: Assuming critical point = extremum f(x) = $x^3$ has f'(0) = 0 but no extremum at x = 0. Must verify sign change.

Mistake 3: Second derivative test when f''(c) = 0 f(x) = $x^4$ at x = 0: f'(0) = 0, f''(0) = 0 (INCONCLUSIVE). Must use first derivative test.

Mistake 4: Using MVT on wrong intervals MVT requires f to be differentiable on the OPEN interval and continuous on the CLOSED interval. If f has a non-differentiable point inside, MVT doesn't apply.

Mistake 5: Optimization without checking domain In the open box problem, x must be between 0 and a/2. The critical point x = a/6 is in this range, but x = a/2 is not valid (gives zero volume).

Mistake 6: Sign error in normal slope Normal slope = -1/f'(x0), not 1/f'(x0). The negative sign is essential.