We have been finding a set of optimal parameters for the linear least-squares regression minimization problem by identifying critical points, i. e. points at which the gradient of a function is the zero vector, of the following function: F(αo, ... ,αm) = ΣNn = 1(α0, + α1xn1 + ... + αMxnM - yn)2
Please help to justify this methodology in the following way. Letting G: Rd - R be any function that is differentiable everywhere, show that, if G has a local minimum at a point xo, then its gradient is the zero vector there, i. e., ΔG(x0) = 0.