Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem
If the functions \(f,g:I\rightarrow \mathbb{R}\) are differentiable on the interval \(I\subseteq \mathbb{R}\), \(a\in I,\) then there exists a function \(\bar{c}:I\rightarrow I\) such that $$ \left[ f\left( x\right) -f\left( a\right) \right] g^{\left( 1\right) }\left( \bar{c}\left( x\right...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Publishing House of the Romanian Academy
2015-12-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://www.ictp.acad.ro/jnaat/journal/article/view/1056 |
| _version_ | 1818199366198886400 |
|---|---|
| author | Beatrix-Mihaela Pop Dorel Duca |
| author_facet | Beatrix-Mihaela Pop Dorel Duca |
| author_sort | Beatrix-Mihaela Pop |
| collection | DOAJ |
| description |
If the functions \(f,g:I\rightarrow \mathbb{R}\) are differentiable on the
interval \(I\subseteq \mathbb{R}\), \(a\in I,\) then there exists a function \(\bar{c}:I\rightarrow I\) such that
$$
\left[ f\left( x\right) -f\left( a\right) \right] g^{\left( 1\right) }\left(
\bar{c}\left( x\right) \right) =\left[ g\left( x\right) -g\left(
a\right) \right] f^{\left( 1\right) }\left( \bar{c}\left( x\right)
\right) ,\text{ for }x\in I.
$$
In this paper we study the differentiability of the function \(\bar{c}\),
when
$$
f^{\left( k\right) }\left( a\right) g^{\left( 1\right) }\left( a\right)
=f^{\left( 1\right) }\left( a\right) g^{\left( k\right) }\left( a\right) ,
\text{ for all }k\in \{1,...,n-1\}
$$
and
$$
f^{\left( n\right) }\left( a\right) g^{\left( 1\right) }\left( a\right) \neq
f^{\left( 1\right) }\left( a\right) g^{\left( n\right) }\left( a\right).
$$
|
| first_indexed | 2024-12-12T02:20:37Z |
| format | Article |
| id | doaj.art-a3771e81a2df4d6088a7e1a8d49175da |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-12-12T02:20:37Z |
| publishDate | 2015-12-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-a3771e81a2df4d6088a7e1a8d49175da2022-12-22T00:41:41ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2015-12-01441Second order differentiability of the intermediate-point function in Cauchy's mean-value theoremBeatrix-Mihaela Pop0Dorel Duca1Babeş-Bolyai UniversityBabeş-Bolyai University If the functions \(f,g:I\rightarrow \mathbb{R}\) are differentiable on the interval \(I\subseteq \mathbb{R}\), \(a\in I,\) then there exists a function \(\bar{c}:I\rightarrow I\) such that $$ \left[ f\left( x\right) -f\left( a\right) \right] g^{\left( 1\right) }\left( \bar{c}\left( x\right) \right) =\left[ g\left( x\right) -g\left( a\right) \right] f^{\left( 1\right) }\left( \bar{c}\left( x\right) \right) ,\text{ for }x\in I. $$ In this paper we study the differentiability of the function \(\bar{c}\), when $$ f^{\left( k\right) }\left( a\right) g^{\left( 1\right) }\left( a\right) =f^{\left( 1\right) }\left( a\right) g^{\left( k\right) }\left( a\right) , \text{ for all }k\in \{1,...,n-1\} $$ and $$ f^{\left( n\right) }\left( a\right) g^{\left( 1\right) }\left( a\right) \neq f^{\left( 1\right) }\left( a\right) g^{\left( n\right) }\left( a\right). $$ https://www.ictp.acad.ro/jnaat/journal/article/view/1056Cauchy theoremintermediate pointmean-value theorem |
| spellingShingle | Beatrix-Mihaela Pop Dorel Duca Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem Journal of Numerical Analysis and Approximation Theory Cauchy theorem intermediate point mean-value theorem |
| title | Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem |
| title_full | Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem |
| title_fullStr | Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem |
| title_full_unstemmed | Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem |
| title_short | Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem |
| title_sort | second order differentiability of the intermediate point function in cauchy s mean value theorem |
| topic | Cauchy theorem intermediate point mean-value theorem |
| url | https://www.ictp.acad.ro/jnaat/journal/article/view/1056 |
| work_keys_str_mv | AT beatrixmihaelapop secondorderdifferentiabilityoftheintermediatepointfunctionincauchysmeanvaluetheorem AT dorelduca secondorderdifferentiabilityoftheintermediatepointfunctionincauchysmeanvaluetheorem |