Some mean value theorems as consequences of the Darboux property
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. Th...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2017-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/142/2/mb142_2_9.pdf |
| _version_ | 1818135495105839104 |
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| author | Dan Ştefan Marinescu Mihai Monea |
| author_facet | Dan Ştefan Marinescu Mihai Monea |
| author_sort | Dan Ştefan Marinescu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T09:25:25Z |
| format | Article |
| id | doaj.art-aa106337cd31484eb4fe93edcbba5b90 |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-11T09:25:25Z |
| publishDate | 2017-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-aa106337cd31484eb4fe93edcbba5b902022-12-22T01:13:10ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362017-07-01142221122410.21136/MB.2016.0032-15MB.2016.0032-15Some mean value theorems as consequences of the Darboux propertyDan Ştefan MarinescuMihai MoneaThe 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.http://mb.math.cas.cz/full/142/2/mb142_2_9.pdf Darboux function mean value theorem continuous function integrable function differentiable function arithmetic mean geometric mean harmonic mean |
| spellingShingle | Dan Ştefan Marinescu Mihai Monea Some mean value theorems as consequences of the Darboux property Mathematica Bohemica Darboux function mean value theorem continuous function integrable function differentiable function arithmetic mean geometric mean harmonic mean |
| title | Some mean value theorems as consequences of the Darboux property |
| title_full | Some mean value theorems as consequences of the Darboux property |
| title_fullStr | Some mean value theorems as consequences of the Darboux property |
| title_full_unstemmed | Some mean value theorems as consequences of the Darboux property |
| title_short | Some mean value theorems as consequences of the Darboux property |
| title_sort | some mean value theorems as consequences of the darboux property |
| topic | Darboux function mean value theorem continuous function integrable function differentiable function arithmetic mean geometric mean harmonic mean |
| url | http://mb.math.cas.cz/full/142/2/mb142_2_9.pdf |
| work_keys_str_mv | AT danstefanmarinescu somemeanvaluetheoremsasconsequencesofthedarbouxproperty AT mihaimonea somemeanvaluetheoremsasconsequencesofthedarbouxproperty |