The Lagrange multiplier rule revisited

In this paper we give a short novel proof of the well-known Lagrange multiplier rule, discuss the sources of the power of this rule and consider several applications of this rule. The new proof does not use the implicit function theorem and combines the advantages of two of the most well-known proofs: it provides the useful geometric insight of the elimination approach based on differentiable curves and technically it is not more complicated than the simple penalty approach. Then we emphasize that the power of the rule is the reversal of order of the natural tasks, elimination and differentiation. This turns the hardest task, elimination, from a nonlinear problem into a linear one. This phenomenon is illustrated by several convincing examples of applications of the rule to various areas. Finally we give three hints on the use of the rule.