The Arithmetic-Geometric Mean Inequality Revisited: Elementary Calculus and Negative Numbers

Abstract

Just recently, in teaching theory of interest to actuarial students, I ran into a problem that used the long-known result that the arithmetic mean of n nonnegative numbers exceeds their geometric mean, except when all n numbers are identical in which case the two means coincide. [1, p. 39]. So I decided to produce a proof suitable for first year (i.e., one-variable) calculus students which appears to be new or at least little known. The other proofs in the literature require no calculus at all or, on the contrary, some understanding of advanced calculus. The method produces various corollaries which I did not note in the literature. This in turn inspires a brief discussion of what becomes of the inequality when nonnegativity requirements are relaxed-a point rarely if ever treated heretofore. Let xl, x2,..., xn be n arbitrary nonnegative numbers. We wish to show