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Characterizing zero-derivative points

  • Sanjo ZlobecEmail author
Article

Abstract

We study smooth functions in several variables with a Lipschitz derivative. It is shown that these functions have the “envelope property”: Around zero-derivative points, and only around such points, the functions are envelopes of a quadratic parabolloid. The property is used to reformulate Fermat’s extreme value theorem and the theorem of Lagrange under slightly more restrictive assumptions but without the derivatives.

Keywords

Zero-derivative point Fermat’s extreme value theorem Theorem of Lagrange 

Mathematics Subject Classification (2000)

26B05 90C30 

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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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