On Some Cauchy Type Mean-Value Theorems with Applications

Some Cauchy-type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule, and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials, and some trigonometric functions. Further, we obtain some exponential, logarithmic, and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.