The method of Lagrange multipliers is one of the milestones of Mathematical Analysis (Calculus), not only because of its intrinsic, historical significance both in theoretical as well as in practical matters, but also because it has been of almost excluding use when trying to determine and analyze constrained local extrema (see, for example, [1], [2], [3], [6], [7], [8] and further references therein).
In [9], F. Zizza described two alternative methods which eliminate the multipliers from the first derivative test, i.e., from that part of the calculation in which the critical points are determined, by using differential forms. In addition, and by the further use of a computer program, he showed experimentally that this method is faster than the previous one He did not indicate however a corresponding, alternative method for analyzing the critical points with regards to the possibilities of being local maxima, minima, or saddle type.
In [4] we developed a method which allows to make that analysis, at least for the limited cases in which the second derivative test is feasible to be of use, by also eliminating at the same time the multipliers
In [5] we outlined a comprehensive method, without Lagrange Multipliers, which woks even in those cases where the second derivative test fails.
It shall be the purpose of this workshop to present that method for determining and analyzing constrained local extrema which, as previously stated, represents a different alternative to all previous works on the topic: no use of Lagrange Multipliers and/or differential forms is made in it.
It should be emphasized that this method is not only very effective and straightforward, but also very simple in its conception and in the requirements of theoretical background material. In fact, in order to be performed correctly by the students, they only need to have a basic knowledge of three very central results from Calculus: the Implicit Function Theorem, the Chain Rule and, in some cases, the Taylor´s Formula.
We shall include, as part of our exposition, carefully chosen examples where a step by step comparison is established with the previously mentioned methods. Thus, the target audience is composed by those educators and students interested in learning an alternative, computationally faster method to treat the problem. With only those mentioned limitations, the number of participants shall also depend on the facilities available at the event. We shall require the use of cannon and blackboard.

References
[1] T.M. Apostol, Calculus, Volume II, Blaisdell, 1967.
[2] C. Caratheodory, Calculus of Variations and Partial Differential Equations II, Holden-Day, 1965.
[3] W.H. Fleming, Functions of Several Variables, Addison-Weslley, 1965.
[4] S. Gigena, M. Binia, D. Abud, Extremos Condicionados: Una propuesta metodológica para su resolución (Constrained Extrema: A Methodological Proposal for its Resolution), Revista de Educación Matemática, Vol. 16, Nº 3, (2001), 31-53.
[5] S. Gigena, Extremos Condicionados sin Multiplicadores de Lagrange, (Constrained Extrema without Lagrange Multipliers). Acta Latino-americana de Matemática Educativa, (2006), 150-155.
[6] J. Marsden, J. and A.J. Tromba, Vector Calculus, 2nd. edition, Freeman, 1976.
[7] Y. Murata, Mathematics for Stability and Optimization of Economic Systems,
Academic Press, 1977.
[8] R.E. Williamson, R.H. Crowell and H.F. Trotter, Calculus of Vector Functions, 3rd. edition, Prentice Hall, 1972.
[9] F. Zizza, Differential Forms for Constrained Max-Min Problems: Eliminating
Lagrange Multipliers, The College Math Journal, Vol. 29 – Nº. 5, (1998), 387-396.