Riddles of Rolle ’ S Theorem

" I think you might do something better with the time " , Alice said, " than wasting it in asking riddles that have no answer ... " Getting started First year undergraduates usually learn about the classical Rolle's theorem saying that between two consecutive zeros of a smooth function f one can always find at least one zero of its derivative f ′. In this paper we study a generalization of Rolle's theorem dealing with the zeros of higher derivatives for a class of smooth functions which we call n-nice. For such functions we obtain some mysteriously looking additional inequalities governing the mutual arrangement of the zeros of different derivatives. The considered topic is so classical that we find it impossible to be sure that our results are new. However, we were unable to trace anything similar in the literature although such facts might have been known to at least Chevalier Augustin Cauchy if not to Sir Isaac Newton himself. Consider a smooth function f that on a certain interval has n distinct real zeros which we denote by x (0)