Objective function, Differentiability, The second order

This lecture explains how first derivatives give information about the slope of a function, while second derivatives provide information about its curvature. The Hessian matrix, or second derivatives matrix, is introduced, showing how it characterizes the convexity of a function. The properties of the Hessian matrix, such as symmetry and positive semidefiniteness, are discussed in the context of function convexity. The lecture also covers reducing problems to one dimension, defining uni-dimensional functions, and interpreting directional curvature. The relationship between second derivatives and function curvature is explored, emphasizing the role of the Hessian matrix in determining function convexity.