A Note on the Generalization of the Mean Value Theorem

In this paper, a new generalization of the mean value theorem is firstly established. Based on the Rolle's theorem, a simple proof is provided to guarantee the correctness of such a generalization. Some corollaries are evidently obtained by the main result. It will be shown that the mean value theorem, the Cauchy's mean value theorem, and the mean value theorem for integrals are the special cases of such a generalized form. We can simultaneously obtain the upper and lower bounds of certain integral formulas and verify inequalities by using the main theorems. Finally, two examples are offered to illustrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Note on the Generalization of the Mean Value Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38258.pdf Paper URL : https://www.ijtsrd.com/mathemetics/calculus/38258/a-note-on-the-generalization-of-the-mean-value-theorem/yeongjeu-sun