Practical Class 7. Conditional Extremum

This practical class is designed to help students master the method of Lagrange multipliers for solving conditional extremum problemsa key topic in mathematical analysis with broad applications in economics, geometry, and the natural sciences.

The session begins with a brief theoretical review, including the geometric interpretation of a conditional extremum and the step-by-step Lagrange multiplier algorithm. The core of the class is dedicated to hands-on problem-solving, with tasks organized into thematic blocks. Students will work through simple linear constraints, more challenging nonlinear constraints, and problems involving functions of three variables.

Applied problems are also featured, such as maximizing a firms profit under resource limits and finding the optimal dimensions of a geometric solid. The class concludes with a short independent assignment to assess understanding, followed by a detailed homework section with solving tips and an overview of common mistakes to avoid