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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?
Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Higher Education
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A