We derive an extremum principle. It can be treated as an intermediate result between
the celebrated smooth-convex extremum principle due to Ioffe and Tikhomirov and the
Dubovitskii–Milyutin theorem. The proof of this principle is based on a simple generalization of the
Fermat’s theorem, the smooth-convex extremum principle and the local implicit function theorem.
An integro-differential example illustrating the new principle is presented.