THE EXTENSION AND APPLICATION OF THE DIFFERENTIAL MEAN-VALUE THEOREM

The Lagrange mean-value theorem and the Cauchy mean-value theorem are extensions of the Rolle mean-value theorem.In this article,the Rolle mean-value theorem has been concluded and deduced in few more forms that helped to expand the use of the Rolle mean-value theorem.Also,the article has demonstrated of the application of differential mean-value theorem in derivative limit,derivative estimate value,existence of root of an equation,proof of inequality and calculation of functional limit upon many examples.