  Two are the main objectives of this article: first, we introduce a method for
determining and analyzing constrained local extrema that provides a different
alternative to all previous works on the topic, by eliminating Lagrange
multipliers and reducing constrained problems to unconstrained ones; Second, we
also develop another method for analyzing and determining the nature of known
critical points by resorting, when needed, to higher order derivatives. The
result presented here constitutes our proposal of solution to a long standing
problem: that of extending to higher dimensions what has been known for many
years in the case of functions defined on open sets of the real line, regarding
the classification of critical points when enough derivatives are also known at
those points.
