Means and the mean value theorem

Authors: Merikoski, Jorma¹; Halmetoja, Markku²; Tossavainen, Timo³

Source: International Journal of Mathematical Education, Volume 40, Number 6, January 2009 , pp. 729-740(12)

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Abstract:

Let I be a real interval. We call a continuous function μ : I × I → a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → 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

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00207390902825328

Affiliations: 1: Department of Mathematics and Statistics, University of Tampere, Finland 2: Mantta Senior High School, FI-35801 Mantta, Finland 3: Department of Teacher Education, University of Joensuu, FI-57101 Savonlinna, Finland

Publication date: 2009-01-01

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