In this paper, using different geometric means of the introduction of the corresponding auxiliary function, the proof of Lagrange's theorem Secondly, the Lagrange mean value theorem in some proof of identity and inequality in a wide range of Example of typical article classified research, and a simple analysis of a number of topics and problem-solving skills in problem-solving Third, the use of Lagrange's theorem proving the points of the study on some conclusions that theorem in the Lagrangian points in the study also has a wide range of Finally discusses the Lagrange mean value theorem proof method of constructing auxiliary function in order to expand on the idea of theorem