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Keywords:
Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function
Lagrange mean-value theorem; mean; Darboux property of derivative; vector-valued function
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.
References:
[1] Berrone, L. R., Moro, J.: Lagrangian means. Aequationes Math. 55 (1998), 217-226. DOI 10.1007/s000100050031 | MR 1615392 | Zbl 0903.39006
[2] Matkowski, J.: Mean value property and associated functional equation. Aequationes Math. 58 (1999), 46-59. DOI 10.1007/s000100050006 | MR 1714318
[3] Matkowski, J.: A mean-value theorem and its applications. J. Math. Anal. Appl. 373 (2011), 227-234. DOI 10.1016/j.jmaa.2010.06.057 | MR 2684472 | Zbl 1206.26032
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