1.由柯西不等式:
(y+z+z+x+x+y)(x^2/(y+z)+y^2/(z+x)+z^2/(x+y))>=(x+y+z)^2
上式整理得:x^2/(y+z)+y^2/(z+x)+z^2/(x+y)>=(x+y+z)/2
再由均值不等式x+y+z>=3*三次根号(xyz)=3
所以x^2/(y+z)+y^2/(z+x)+z^2/(x+y)>=(x+y+z)/2>=3/2
所以x^2/(y+z)+y^2/(z+x)+z^2/(x+y)最小值为3/2,当x=y=z=1的时候取到.
2.证明:
先考虑证明:
x/(x+1)-1/4