Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem

Authors

  • Beatrix-Mihaela Pop Babeş-Bolyai University, Romania
  • Dorel Duca Babeş-Bolyai University, Romania

DOI:

https://doi.org/10.33993/jnaat441-1056

Keywords:

Cauchy theorem, intermediate point, mean-value theorem
Abstract views: 262

Abstract

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
$$ f^{\left( n\right) }\left( a\right) g^{\left( 1\right) }\left( a\right) \neq f^{\left( 1\right) }\left( a\right) g^{\left( n\right) }\left( a\right). $$

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References

D.I. Duca and O.T. Pop, Concerning the intermediate point in the mean value theorem, Mathematical Inequalities & Applications, 12 (2009), no. 3, pp. 499-512. http://doi.org/10.7153/mia-12-38 DOI: https://doi.org/10.7153/mia-12-38

D.I. Duca and O. Pop, On the intermediate point in Cauchy’s mean-value theorem, Mathematical Inequalities & Applications, 9 (2006), no. 3, pp. 375-389. http://doi.org/10.7153/mia-09-37 DOI: https://doi.org/10.7153/mia-09-37

B.-M. Pop and D.I. Duca, The derivability of the intermediate point function in Cauchy’s mean-value theorem, Didactica Mathematica, 32 (2014), pp. 87-100 (in Romanian). DOI: https://doi.org/10.33993/jnaat441-1056

T. Trif, Asymptotic behavior of intermediate point in certain mean value theorems, J. Math. Ineq., 2 (2008), pp. 151-161. http://doi.org/10.7153/jmi-02-15 DOI: https://doi.org/10.7153/jmi-02-15

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Published

2015-12-18

How to Cite

Pop, B.-M., & Duca, D. (2015). Second order differentiability of the intermediate-point function in Cauchy’s mean-value theorem. J. Numer. Anal. Approx. Theory, 44(1), 100–109. https://doi.org/10.33993/jnaat441-1056

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