The book contains about 700 problems for finding extremes for a finite-dimensional case, for problems of classical variational calculus, optimal control and convex programming. Contains elements of functional analysis, differential calculus and convex analysis. The book provides the theory needed to solve problems, and examples. The basic solution of all problems is a single principle, dating back to Lagrange .. Some of the tasks are given with solutions. There are a large number of difficult tasks that can be used as term papers and diploma papers . For university students in the specialties "Mathematics" and "Applied Mathematics", as well as for graduate students and researchers . Recommended by the Educational and Methodological Council for Mathematics and Mechanics of the UMO for classical university education as a target for students of higher educational institutions studying in a group of mathematical directions and specialties .