Geometric approach to finding the conditional extrema

In this paper, we give a geometric interpretation and a geometric proof of the necessary condition for the existence of a constrained extremum. The presented approach can be applied to finding constrained extrema of nondifferentiable functions (i.e., when Lagrange's method of undetermined multipliers is not applicable in the “classical” form). The following examples are considered: the inequality of arithmetic and geometric means, Young's inequality for products, and Jensen's inequality.