Characterizing zero-derivative points

  • Sanjo ZlobecEmail author


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.


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

Mathematics Subject Classification (2000)

26B05 90C30 


  1. 1.
    Androulakis I.P., Maranas C.D., Floudas C.A.: αBB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)CrossRefGoogle Scholar
  2. 2.
    Adjiman C.S., Floudas C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9, 23–40 (1996)CrossRefGoogle Scholar
  3. 3.
    Apostol T.M.: Calculus, vol. II: Calculus of Several Variables with Applications to Probability and Vector Analysis. Blaisdell, New York (1965)Google Scholar
  4. 4.
    Stewart J.: Single Variable Calculus. Brooks/Cole, Pacific Grove (1987)Google Scholar
  5. 5.
    Zlobec S.: The fundamental theorem of calculus for Lipschitz functions. Math. Commun. 13, 215–232 (2008)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

Personalised recommendations