  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.
