Mathematica Bohemica, Vol. 142, No. 2, pp. 211-224, 2017

Some mean value theorems as consequences of the Darboux property

Dan Ştefan Marinescu, Mihai Monea

Received June 24, 2015.  First published December 16, 2016.

Abstract:  The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.
Keywords:  Darboux function; mean value theorem; continuous function; integrable function; differentiable function; arithmetic mean; geometric mean; harmonic mean
Classification MSC:  26A15, 26A24, 26A42
DOI:  10.21136/MB.2016.0032-15

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Affiliations:   Dan Ştefan Marinescu, National College "Iancu de Hunedoara", Strada Libertăţii No. 2, Hunedoara, 331032 Romania, e-mail:;
Mihai Monea, Polytechnic University, Bucharest & National College "Decebal", Str. Decebal, Bl. 8, Deva, 330012 Romania, e-mail:

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