D.S. Telyakovskii, S.A. Telyakovskii. Geometric approach to finding the conditional extrema ... P. 244-254

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.

Keywords: constrained extremum, level surface, Lagrange multipliers.

Received January 9, 2020

Revised October 7, 2020

Accepted October 26, 2020

Dmitry Sergeevich Telyakovskii, Cand. Sci. (Phys.-Math.), Prof., National Research Nuclear University (MEPhI), Moscow, 115409 Russia, e-mail: dtelyakov@mail.ru

Sergey Alexandrovich Telyakovskii, Dr. Phys.-Math. Sci., Prof., Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia, e-mail: sergeyaltel@yandex.ru

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Cite this article as: D.S. Telyakovskii, S.A. Telyakovskii. Geometric approach to finding the conditional extrema, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 244–254.