Means and the mean value theorem

• Autores: Jorma K. Merikoski, Markku Halmetoja, Timo Tossavainen
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 40, Nº. 6, 2009, págs. 729-740
• Idioma: inglés
• DOI: 10.1080/00207390902825328
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• Resumen

  • 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 ? between a and b such that f(b) - f(a) = f '(?)(b - a). We study under what conditions ? 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?