GENERALIZATIONS OF LAGRANGE AND CAUCHY MEAN-VALUE THEOREMS

Abstract

Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented. 1. Introduction Recall that a function M : I ! I is called a mean in a nontrivial interval I R I if it is internal, that is if min(x; y) M(x; y) max(x; y) for all x; y 2 I: The mean M is called strict if these inequalities are strict for all x; y 2 I; x 6= y; and symmetric if M(x; y) =M(y; x) for all x; y 2 I: The Lagrange Mean-Value Theorem can be formulated in the following way. If a function f : I ! R is di¤erentiable, then there exists a strict symmetric mean L : I ! I such that, for all x; y 2 I; x 6= y; (1) f(x) f(y) x y = f 0(L(x; y)): If f 0 is one-to-one then, obviously, L ] := L is uniquely determined and is called a Lagrange mean generated by f: Note that formula (1) can be written in the form 1 2 f(x) f( 2 ) x x+y 2 + f( 2 ) f(y) x+y 2 y ! = f 0(L(x; y)); x; y 2 I; x 6= y; or, setting A(x; y) := x+y 2 , in the form (2) A f(x) f(A(x; y)) x A(x; y) ; f(A(x; y)) f(y) A(x; y) y = f 0(L(x; y)); x; y 2 I; x 6= y; which shows a relationship of the mean-value theorem and the arithmetic mean. This equation has the following geometrical interpretation. The arithmetic mean of the slope of chord of the graph of f passing through the points (x; f(x)) and x+y 2 ; f x+y 2 and the slope of chord of the graph passing through the points 2000 Mathematics Subject Classi cation. Primary 26A24.