Mean-value theorem for vector-valued functions

For a differentiable function f I→ Rᵏ, where I is a real interval and k∈ N, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean M I²→ I such thatf(x)- f( y) =( x-y) f'( M(x,y)) , x,y∈ I, are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.