  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'.
