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Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function
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.
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