Geodesic
Mean-value theorem for vector-valued functions
Mathematica Bohemica, Tome 137 (2012) no. 4, pp. 415-423.

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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.
DOI : 10.21136/MB.2012.142997
Classification : 26A24, 26E60
Keywords: Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function 
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Matkowski, Janusz. Mean-value theorem for vector-valued functions. Mathematica Bohemica, Tome 137 (2012) no. 4, pp. 415-423. doi : 10.21136/MB.2012.142997. http://geodesic.mathdoc.fr/articles/10.21136/MB.2012.142997/

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