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1·In this paper, using different geometric means, the introduction of the corresponding auxiliary function of the Lagrange theorem proof explored.

本文首先采用不同的几何手段，引进相应的辅助函数，对拉格朗日定理的证明进行了探索。

2·On the basis of these theories, Rolle mean value theorem, Lagrange mean value theorem and Cauchy mean value theorem are proved by constructing nested interval.

在此基础上通过构造区间套依次证明了罗尔中值定理、拉格朗日中值定理和柯西中值定理。

3·This paper gives the new method to prove the cauchy mean value theorem which also may be deduced from the Lagrange mean value theorem.

给出柯西中值定理的一个新的证法，说明柯西中值定理也可由拉格朗日中值定理导出。

4·Furthermore, utilizing the Lagrange method of multipliers and the implicit theorem to work out the critical value which makes one of those inequality locally inverted.

然后利用拉格朗日乘数法与隐函数定理，求出了使其中一不等式局部反向的临界值。

5·Secondly, the Lagrange mean value theorem in some proof of identity and the inequality in a wide range of applications.

其次，拉格朗日中值定理在一些等式和不等式的证明中应用十分广泛。

6·The paper makes an analysis and inquiry about the differences among the Roue Theorem. Lagrange Thoorem and Cauchy Theorem.

本文就罗尔定理、拉格朗日定理和柯西定理三者的区别与联系作了分析与探讨。

7·We establish a Lagrange multiplier theorem for strict efficiency in convex settings and express strict points as saddle points of an appropriate Lagrangian function.

讨论凸多目标最优化问题的严有效解，建立了拉格朗日乘子定理，并把严有效解表示为一个适当的拉格朗日函数的鞍。

8·The paper sums up the application of Lagrange mean theorem in five aspects in high mathematics, and give an example to illustrate its application.

总结了高等数学中拉格朗日中值定理五个方面的应用，并举例加以说明。

9·Recent trends in the Lagrange equation's conversion and form from inertial system to non-inertial system, the application of energy theorem, energy conservation law, etc. are introduced.

介绍了在非惯性系中建立动力学方程的方法，惯性系中拉格朗日方程在非惯性系中的转换形式，以及非惯性系中的能量定理和能量守恒定律的应用等研究成果。

10·Finally discusses the Lagrange mean value theorem proof method of constructing auxiliary function in order to expand on the idea of theorem proving.

最后探讨了拉格朗日中值定理证明中辅助函数的构造方法，以此拓展对定理证明的思路。