Implicit Functions: Extrema and Lagrange Multipliers

This lecture covers the concept of extrema under constraints, focusing on determining local extrema of a function under the constraint g(x, y) = 0. The method of Lagrange multipliers is introduced as a necessary condition for the function to have a local extremum. The lecture explains the application of Lagrange multipliers in finding extrema and introduces the Lagrangian of the minimization problem under constraints. Generalization to functions of multiple variables is discussed, highlighting the conditions for local extrema. Various mathematical derivations and proofs are presented to illustrate the application of Lagrange multipliers in optimization problems.