Characterizing zero-derivative points

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

• Type: Journal
• Year: 2010
• Where: JGO
• Authors: Sanjo Zlobec
• Keywords: Fermat’s Extreme Value, JGO 2010, Mathematics Subject Classification, Zero-derivative Point