On Use of Entropy Function for Validating Differential Calculus Results

Rolle's Theorem (RT) and Lagrange's Mean-value Theorem (LMVT) are significant for pure and applied mathematics, and they have applications in various other fields such as management, physics etc. RT is significant in finding the projectile trajectory's maximum height and in information theory, and the entropy function (measure) is used to measure the uncertainty of information.RT is used to analyze the graphs of annual performance in any field.Since information is necessary to analyze any performance and in information theory, entropy measure is a significant tool to quantize the uncertainty so by using the concept of RT and LMVT in information theory the uncertainty and vagueness or noise can be minimized or maximized.In this manuscript, the concept of differential calculus, i.e., RT and LMVT are used for validation of the entropy function.In this paper, characteristics of differential calculus in information entropy function have been discussed.It has been shown that the entropy function satisfies RT and LMVT.It also describes the conditions when Rolle's Theorem becomes the necessary and sufficient condition for entropy function.Theorems are proved related to the concept of differential calculus in information theory which shows that by using the existing entropy function some new entropies can be derived.