Implicit Function Theorem

This lecture covers the generalization of the Implicit Function Theorem for functions in Ck(U) for U⊆R², where a unique function g is defined locally around a point (a₁, a₂, ..., an) such that f(x, g(x)) = 0. It explores the concept of supporting hyperplanes, local extrema, and the calculation of higher-order derivatives. The lecture concludes with an analysis of stationary points and the classification of local maxima, minima, and saddle points based on the Hessian matrix eigenvalues.