A global theory of flexes of periodic functions

For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $\phi_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on $S^1$ if and only if $\phi_s\ge u$ (resp. $\phi_s\le u$) and the preimage $(\phi-u)^{-1}(0)$ is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.