Implicit Functions: Lagrange Multipliers

This lecture covers the concept of implicit functions and Lagrange multipliers. It starts by introducing the implicit function theorem for a function f: R^2 -> R, where f is continuously differentiable. The theorem states that if f(a, b) = 0 for some point (a, b), then there exists a function y(x) defined in a neighborhood of a such that (x, y(x)) satisfies f(x, y(x)) = 0. The demonstration shows the existence of suitable intervals for x and y, ensuring the monotonicity of the function. Finally, it concludes by proving that the function ga is strictly monotone at the point (a, b) and ga(b) = 0.