The article provides a detailed presentation of algorithms for finding results obtained by J.L. Lagrange. The theory of extrema of functions of many variables, as a part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems about the conditional extremum of a function of many variables. The relevance of this topic is determined by the fact that when solving modern problems, methods of solving problems for extremum functions of many variables, obtained in the middle of the 18th and the beginning of the 20 th centuries, are used. A special place here is occupied by L. Eiler and J.L. Lagrange. The purpose of the article is to show the algorithm for finding the conditional and local extremum obtained by J.L. Lagrange.