12. Prove the arithmetic-geometric means inequality. If xib0 for all i, 1 n Xn i1 xib Yn...

12. Prove the arithmetic-geometric means inequality. If xib0 for all i, 1 n

Xn i¼1 xib Yn i¼1 xi

!1=n

:

(Hint: The result is apparently true if some xi ¼ 0, so assume all xi > 0. Take logarithms and consider if ln x is a concave or convex function.)

Remark 9.165 When fxig are both positive and negative, this inequality is satisfied with the collection, fjxijg.