只須求出1/(x+y)+1/(y+z)+1/(z+x)的最大值即可知道k的范圍.
∵x+y+z=xyz
∴1/(xy)+1/(yz)+1/(zx)=1
由柯西不等式知：
[1/(x+y)+1/(y+z)+1/(z+x)]^2
≤(1^2+1^2+1^2)[1/(x+y)^2+1/(y+z)^2+1/(z+x)^2]
而且(x+y)^2=x^2+y^2+2xy≥2xy+2xy=4xy
∴1/(x+y)^2≤1/(4xy)
同理：1/(y+z)^2≤1/(4yz),1/(z+x)^2≤1/(4zx)
∴1/(x+y)^2+1/(y+z)^2+1/(z+x)^2≤[1/(xy)+1/(yz)+1/(zx)]/4=1/4
∴[1/(x+y)+1/(y+z)+1/(z+x)]^2
≤(1^2+1^2+1^2)[1/(x+y)^2+1/(y+z)^2+1/(z+x)^2]
≤3/4
∴1/(x+y)+1/(y+z)+1/(z+x)≤√3/2
以上不等式等號成立的條件是x=y=z=√3
即1/(x+y)+1/(y+z)+1/(z+x)最大值為√3/2
∴當(dāng)k≥√3/2時(shí),1/(x+y)+1/(y+z)+1/(z+x)≤k恒成立
k∈(√3/2 ,+∞)