Means and the mean value theorem

Abstract

Let I be a real interval. We call a continuous function μ : I × I → ℝ a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → ℝ be a differentiable and strictly convex or strictly concave function. If a, b ∈ I with a ≠ b, then there exists a unique number ξ between a and b such that f(b) − f(a) = f ′(ξ)(b − a). We study under what conditions ξ is a proper mean of a and b, and what kind of means are obtained by applying certain f 's. We also study the converse problem: Given a proper mean μ(a, b), does there exist f such that f(b) − f(a) = f ′(μ(a, b))(b − a) for all a, b ∈ I with a ≠ b?

Keywords:

• mean
• mean value theorem
• teaching of mathematics
• undergraduate mathematics

Acknowledgements

We thank Jyrki Lahtonen of University of Turku and Tuomas Korppi of University of Helsinki for valuable suggestions.