MATHEMATICA BOHEMICA, Vol. 137, No. 4, pp. 415-423, 2012

Mean-value theorem for vector-valued functions

Janusz Matkowski

Janusz Matkowski, Faculty of Mathematics Computer Science and Econometric, University of Zielona Góra, ul. Szafrana 4a, PL-65-516 Zielona Góra, Poland, e-mail:

Abstract: For a differentiable function $ f I\rightarrow\mathbb{R}^k,$ where $I$ is a real interval and $k\in\mathbb{N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M I^2\rightarrow I$ such that
f(x)- f( y) =( x-y) f'( M(x,y)) ,\quad x,y\in I,
are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

Keywords: Lagrange mean-value theorem, mean, Darboux property of derivative, vector-valued function

Classification (MSC 2010): 26A24, 26E60

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