On an extended Lagrange claim

Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of fat th at point are nonnegative. That the Lagrange claim is wrong was shown by a c ounterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivative s of a locally Lipschitz function f at a point are positive, then f has a s trict minimum at that point.