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A Few Riddles Behind Rolle’s Theorem

B. Shapiro and M. Shapiro
The American Mathematical Monthly
Vol. 119, No. 9 (November 2012), pp. 787-793
DOI: 10.4169/amer.math.monthly.119.09.787
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Page Count: 7
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A Few Riddles Behind Rolle’s Theorem
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Abstract First year undergraduates usually learn about classical Rolle’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′. In this paper, we study a generalization of Rolle’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 ⊆ ℝa 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.

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