设正实数xyz满足x+2y+z=3则y+z+(x+y)^2

x+2y+z=3
(x+y)+(y+z)=3
[(y+z)+(x+y)²]/[(x+y)(y+z)]
=1/(x+y)+ (x+y)/(y+z)
=(1/3)[3/(x+y)] +(x+y)/(y+z)
=(1/3)[(x+y)+(y+z)]/(x+y) +(x+y)/(y+z)
=1/3 +(1/3)(y+z)/(x+y) +(x+y)/(y+z)
由均值不等式得：
(1/3)(y+z)/(x+y) +(x+y)/(y+z)≥2√(1/3)=2√3/3,当且仅当(y+z)/(x+y)=3(x+y)/(y+z)时取等号
此时[(y+z)+(x+y)²]/[(x+y)(y+z)]有最小值
[(y+z)+(x+y)²]/[(x+y)(y+z)]min=1/3 +2√3/3=(1+2√3)/3
[(y+z)+(x+y)²]/[(x+y)(y+z)]的最小值为(1+2√3)/3
提示：此类题目的数量非常大,究其根本就是运用均值不等式,但需要进行一定的变形,考察的是对均值不等式的理解及运用是否已经熟练掌握.对于生手,确实是有难度的.对于熟练的人,则不成问题.