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A few riddles behind Rolle's theorem
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)In: The American mathematical monthly, ISSN 0002-9890, Vol. 119, no 9, 787-793 p.Article in journal (Refereed) Published
Abstract [en]

First year undergraduates usually learn about classical Rolle'stheorem which says that between two consecutive zeros of a smooth univariate function $f$, one can always find at least one zeroof its derivative $f^\prime$. In this paper, we study a generalization of Rolle's theorem dealing with zeros of higherderivatives of smooth univariate functions enjoying a natural additional property. Namely, we call a smooth function whose $n$th derivative does not vanish on some interval $I\subseteq \bR$ a {\it 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 $n=3$.

Place, publisher, year, edition, pages
2012. Vol. 119, no 9, 787-793 p.
Keyword [en]
Rolle's theorem, polynomial-like functions
National Category
Research subject
URN: urn:nbn:se:su:diva-84473DOI: 10.4169/amer.math.monthly.119.09.787ISI: 000321593500006OAI: diva2:580526
Available from: 2013-01-14 Created: 2012-12-22 Last updated: 2013-08-13Bibliographically approved

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