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Mean-value theorem for vector-valued functions. (English) Zbl 1274.26009
Summary: 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 \[ f(x)-f(y)=(x-y)f'(M(x,y)),\; x,y\in I, \] are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

MSC:
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26E60 Means
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