As one of the core ideas and courses of mathematical analysis, differential calculus plays a pivotal role in the study of functions and is an indispensable part of the composition of mathematical analysis, especially Rolle's Median Theorem, Lagrange's Median Theorem, Cauchy's Median Theorem, and so on, whose universality plays an important role in mathematical research. In this paper, the conditions for the applicability of these theorems are generalised and extended, and the scope of their application is extended from finite closed intervals and open intervals to infinite intervals. Through a series of extension proofs, it is shown that these median theorems still hold in a wider range of intervals. The paper then provides specific applications and explanations of the promoted differential median theorems, demonstrating their convenience and flexibility in solving some specific problems. These generalisations and extensions make the differential median theorems more universal and provide a basis for future higher level mathematical research.

Guo Xiaosong. Generalized application of differential mean value theorem in infinite intervals. Academic Journal of Mathematical Sciences (2024) Vol. 5, Issue 1: 81-88. https://doi.org/10.25236/AJMS.2024.050112.

[1] Department of Mathematics, Fudan University. Mathematical Analysis (Fourth Edition Upper) [M]. Beijing: Higher Education Press, 2017

[2] Q.C. Tang,R. Feng,Z.W. Zhang. An applied study of differential median theorem[J]. Academy, 2021, 14 (13): 13-15.

[3] Department of Mathematics, Tongji University. Higher Mathematics (7th Edition Upper) [M]. Beijing: High Education Press, 2014.

[4] Deng Shao,Yue Xiaorui. A short note on the application of differential median theorem[J]. Science and Technology Innovation and Application, 2019, (20): 184-185+188.

[5] Fan Shoufang. Some problems of the median theorem of calculus [M]. Harbin: Heilongjiang University Press, 2011.