Some mean value theorems as consequences of the Darboux property

Dan Ştefan Marinescu; Mihai Monea

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 2, page 211-224
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Marinescu, Dan Ştefan, and Monea, Mihai. "Some mean value theorems as consequences of the Darboux property." Mathematica Bohemica 142.2 (2017): 211-224. <http://eudml.org/doc/288108>.

@article{Marinescu2017,
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.},
author = {Marinescu, Dan Ştefan, Monea, Mihai},
journal = {Mathematica Bohemica},
keywords = {Darboux function; mean value theorem; continuous function; integrable function; differentiable function; arithmetic mean; geometric mean; harmonic mean},
language = {eng},
number = {2},
pages = {211-224},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some mean value theorems as consequences of the Darboux property},
url = {http://eudml.org/doc/288108},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Marinescu, Dan Ştefan
AU - Monea, Mihai
TI - Some mean value theorems as consequences of the Darboux property
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 2
SP - 211
EP - 224
AB - 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.
LA - eng
KW - Darboux function; mean value theorem; continuous function; integrable function; differentiable function; arithmetic mean; geometric mean; harmonic mean
UR - http://eudml.org/doc/288108
ER -

References

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  1. Chen, Z., 10.2307/3647923, Amer. Math. Monthly 110 (2003), 544-545. (2003) DOI10.2307/3647923
  2. Garcia, T. M., Suarez, P., Solution of problem 1867, Mathematics Magazine 85 (2012), page 153. (2012) Zbl1246.97017MR3324702
  3. Herman, E., Lampakis, E., Witkowski, A., Solution of problem 956, Coll. Math. J. 43 (2012), 338-339. (2012) 
  4. Jarník, V., 10.4064/fm-21-1-48-58, Fundam. Math. 21 (1933), 48-58. (1933) Zbl0007.40102DOI10.4064/fm-21-1-48-58
  5. Kowalewski, G., Interpolation und genäherte Quadratur. Eine Ergänzung zu den Lehr- büchern der Differential- und Integralrechnung, B. G. Teubner, Leipzig und Berlin (1932). (1932) Zbl0004.05605
  6. Marinescu, D. Ş., În legătură cu o problemă de concurs, Recreaţii Matematice 1 (2004), 20-22 (in Romanian). (2004) 
  7. Marinescu, D. Ş., Problem 26546, Gazeta Matematică, seria B 12 (2001). (2001) 
  8. Marinescu, D. Ş., Monea, M., Stroe, M., Teorema lui Jarnik şi unele consecinţe, Revista de Matematică a Elevilor din Timişoara 4 (2010), 3-8 (in Romanian). (2010) 
  9. Orno, P., 10.2307/2689475, Mathematics Magazine 51 (1978), page 245. (1978) MR1572276DOI10.2307/2689475
  10. Pangsriiam, P., Problem 11753, American Mathematical Monthly 121 (2014), page 84. (2014) 
  11. Plaza, A., Rodriguez, C., Problem 1867, Mathematics Magazine 84 (2011), page 150. (2011) 
  12. Precupanu, T., Problem 5.119, Olimpiadele Naţionale de Matematică 1954-2003 (D. Bă- tineţu, I. Tomescu, eds.). Ed. Enciclopedică, Bucureşti (2004), page 146 (in Romanian). 
  13. C. F. Rocca, Jr., A question of integral, Mat: 450 Senior Seminar (2012). (2012) 
  14. Sahoo, P. K., Riedel, T., Mean Value Theorems and Functional Equations, World Scientific, Singapore (1998). (1998) Zbl0980.39015MR1692936
  15. Thong, Duong Viet, Problem 956, Coll. Math. J. 42 (2011), page 329. (2011) 

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