Characterizing zero-derivative points
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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.
KeywordsZero-derivative point Fermat’s extreme value theorem Theorem of Lagrange
Mathematics Subject Classification (2000)26B05 90C30
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