MATH.10:2 - Problem

Changing one value independently can leave the allowed set. A derivative calculated in that direction can then recommend an impossible change. Even with admissible directions, a zero first derivative can describe a maximum, a saddle point or a flat comparison; further reasoning is needed for a minimum.

The reverse difficulty occurs at a constraint boundary: an optimum can have a nonzero derivative because the improving direction is unavailable. A calculation that discards the allowed parameter range loses that conclusion.

These errors share a missing connection between what can vary, what the variation does to the quantity and what the examined family establishes about the original question.