Discussion on the application of Lagrange mean value theorem

Abstract The Lagrange mean value theorem has been widely used in the following aspects; ( 1 )Prove equation; ( 2 )Proof inequality; ( 3 ) Study the properties of derivatives and functions; (4) Prove the conclusion of the mean value theorem; (5) Determine the existence and uniqueness of the roots of the equation; (6 ) Use the mean value theorem to find the limit.