**ERIC Number:**EJ857943

**Record Type:**Journal

**Publication Date:**2009

**Pages:**12

**Abstractor:**As Provided

**Reference Count:**11

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

Means and the Mean Value Theorem

Merikoski, Jorma K.; Halmetoja, Markku; Tossavainen, Timo

International Journal of Mathematical Education in Science and Technology, v40 n6 p729-740 2009

Let I be a real interval. We call a continuous function [mu] : I x I [right arrow] [Bold R] a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I [right arrow] [Bold R} be a differentiable and strictly convex or strictly concave function. If a, b [image omitted] I with a [not equal to] b, then there exists a unique number [Xi] between a and b such that f(b) - f(a) = f '([Xi])(b - a). We study under what conditions [Xi] 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 [mu](a, b), does there exist f such that f(b) - f(a) = f '([mu](a, b))(b - a) for all a, b [element of] I with a [not equal to] b?

Descriptors: Mathematics Instruction, Equations (Mathematics), Problem Solving, Mathematical Logic, Validity, College Mathematics, Undergraduate Study

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A