A Note on the Mean Value Theorems

Por • 26 abr, 2022 • Sección: Ambiente

Kung Kuen Tse

Abstract The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point P on the graph such that the tangent at P is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point Q on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points and Q always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.

Keywords Mean Value TheoremReal Analytic FunctionsIdentity TheoremPower Series

Kung Kuen TseDepartment of Mathematics, Kean University, Union, New Jersey, USA.

DOI: 10.4236/apm.2021.115026   PDF   HTML   XML   120 Downloads   458 Views  

Share and Cite: Tse, K. (2021) A Note on the Mean Value Theorems. Advances in Pure Mathematics11, 395-399. doi: 10.4236/apm.2021.115026.

Advances in Pure Mathematics > Vol.11 No.5, May 2021

https://www.scirp.org/journal/paperinformation.aspx?paperid=109154

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