Some Properties of Smooth Convex Functions and Newton's Method

New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the Hessian matrix is singular at the solution point of the function to be minimized, the gradient of the objective function belongs to the image of its second derivative. Due to this new property of convex functions, Newtonian methods for solving unconstrained optimization problems can be applied without assuming the nonsingularity of the Hessian matrix at the solution of the problem and their rate of convergence in argument can be estimated under fairly general assumptions.