# Article

 Title: Mean-value theorem for vector-valued functions (English) Author: Matkowski, Janusz Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 137 Issue: 4 Year: 2012 Pages: 415-423 Summary lang: English . Category: math . Summary: For a differentiable function ${\bf f}\colon I\rightarrow \mathbb {R}^{k},$ 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 $M\colon I^{2}\rightarrow I$ such that$${\bf f}(x)-{\bf f}( y) =( x-y) {\bf f}'( M(x,y)) ,\quad x,y\in I,$$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented. (English) Keyword: Lagrange mean-value theorem Keyword: mean Keyword: Darboux property of derivative Keyword: vector-valued function MSC: 26A24 MSC: 26E60 idZBL: Zbl 1274.26009 idMR: MR3058273 DOI: 10.21136/MB.2012.142997 . Date available: 2012-11-10T20:29:44Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/142997 . Reference: [1] Berrone, L. R., Moro, J.: Lagrangian means.Aequationes Math. 55 (1998), 217-226. Zbl 0903.39006, MR 1615392, 10.1007/s000100050031 Reference: [2] Matkowski, J.: Mean value property and associated functional equation.Aequationes Math. 58 (1999), 46-59. MR 1714318, 10.1007/s000100050006 Reference: [3] Matkowski, J.: A mean-value theorem and its applications.J. Math. Anal. Appl. 373 (2011), 227-234. Zbl 1206.26032, MR 2684472, 10.1016/j.jmaa.2010.06.057 .

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