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 f at that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally Lipschitz function f at a point are positive, then f has a strict minimum at that point.
- Directional derivatives
- Lipschitz continuous functions
- Minimum points
ASJC Scopus subject areas
- Management Science and Operations Research
- Applied Mathematics
- Control and Optimization