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.
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Zlobec, S. Characterizing zero-derivative points. J Glob Optim 46, 155–161 (2010). https://doi.org/10.1007/s10898-009-9457-4
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DOI: https://doi.org/10.1007/s10898-009-9457-4