For a differentiable function f : I → ℝ k , 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 : I 2 → I such that f (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.