A Few Riddles Behind Rolle%26apos;s Theorem

作者：Shapiro B^*; Shapiro M

来源：American Mathematical Monthly, 2012, 119(9): 787-793.

DOI：10.4169/amer.math.monthly.119.09.787

First year undergraduates usually learn about classical Rolle%26apos;s theorem which says that between two consecutive zeros of a smooth univariate function f, one can always find at least one zero of its derivative f%26apos;. In this paper, we study a generalization of Rolle%26apos;s theorem dealing with zeros of higher derivatives of smooth univariate functions enjoying a natural additional property. Namely, we call a smooth function whose nth derivative does not vanish on some interval I subset of R a polynomial-like function of degree n on I. We conjecture that for polynomial-like functions of degree n with n real distinct roots, there exists a non-trivial system of inequalities completely describing the set of possible locations of their zeros together with their derivatives of order up to n - 1. We describe the corresponding system of inequalities in the simplest non-trivial case n = 3.

• 出版日期2012-11