Lecture 6. Conditional Extremum. Lagrange Multiplier Method

This lecture introduces the concept of conditional extremum the extremum of a function whose variables are linked by additional constraint equations. The geometric interpretation is discussed: on the plane, a level curve of the function touches the constraint curve. The Lagrange multiplier method is presented in detail, including the construction of the Lagrange function, derivation of necessary conditions, and sufficient conditions based on the second differential. Two worked examples are provided: a geometric problem of finding the rectangle of largest area inscribed in an ellipse, and an economic problem of maximizing output under a budget constraint. The lecture concludes with a brief generalization to functions of several variables with multiple constraints. Students gain a practical tool for solving optimization problems with equality constraints.