Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C¹ or C²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C¹ or C²-curves. Surprisingly, what is true for C¹-curves fails to be true in part for C²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.