一道高中的柯西不等式证明已知x1 x2 x3∈R+求证:1/x1+1/x2+/1x3≥2(1/(x1+x2)+1/(x2+x3)+1/(x3+x1))

问题描述:

一道高中的柯西不等式证明
已知x1 x2 x3∈R+
求证:1/x1+1/x2+/1x3≥2(1/(x1+x2)+1/(x2+x3)+1/(x3+x1))

由柯西不等式推广②得
(1/x1)+(1/x2)≥[(1+1)^2/(x1+x2)]
即(1/x1)+(1/x2)≥4/(x1+x2)--*
同理
(1/x3)+(1/x2)≥4/(x3+x2)--&
(1/x1)+(1/x3)≥4/(x1+x3)--#
将#+&+*即得所证式