AbstractFor a real valued periodic smooth functionuonR,n≥ 0, one defines theosculating polynomial φs(of order2n+ 1)at a point s∈Rto be the unique trigonometric polynomial of degreen, whose value and first 2nderivatives atscoincide with those ofuats. We will say that a pointsis aclean maximal flex(resp.clean minimal flex) of the functionuonS1if and only ifφs≥ u(resp.φs≤u) and the preimage (φ - u)-1(0) is connected. We prove thatany smooth periodic function u has at least n+ 1clean maximal flexes

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... of order2n+ 1and at least n+ 1clean minimal flexes of order2n+ 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.