Calculus, constrained minimization and Lagrange multipliers: Is the optimal critical point a local minimizer?

In this short note, we discuss how the optimality conditions for the problem of minimizing a multivariate function subject to equality constraints have been dealt with in undergraduate Calculus. We are particularly interested in the 2 or 3-dimensional cases, which are the most common cases in Calculus courses. Besides giving sufficient conditions to a critical point to be a local minimizer, we also present and discuss counterexamples to some statements encountered in the undergraduate literature on Lagrange Multipliers, such as among the critical points, the ones which have the smallest image (under the function) are minimizers' or a single critical point (which is a local minimizer) is a global minimizer’.