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On the Behaviour of Functions around Zero-Derivative Points

Zero-derivative points are of the fundamental importance in calculus and otimization. It has been recently shown that on intervals around zero-derivative points, and only around zero-derivative points, every smooth function with a Lipschitz derivative is an "envelope" of a parabolloid. In this paper we give two equivalent, but geometrically different, reformulations of this result. They are applied to the three classic theorems: Fermat's extreme value theorem, the mean value theorem, and the Lagrange multiplier theorem. These theorems are augmented over intervals and stated withouth derivatives.

zero-derivative point; Fermat's extreme value theorem; mean value theorem; Lagrange multiplier theorem

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