Question

已知xyz属于R+,x+y+z=1,求证x^3/(y(1-y))+y^3/(z(1-z))+z^3/(x(1-x))大于

Answer 1

x^3/(y(1-y))+y/2+(1-y)/4>=3 三次根号(x^3/(y(1-y))*y/2*(1-y)/4)=3/2 x,

同理

y^3/(z(1-z))+z/2+(1-z)/4>=3/2 y,

z^3/(x(1-x))+x/2+(1-x)/4>=3/2 z,

三个式子一加,

原式>=3/2 (x+y+z)-(x+y+z)/2-(3-x-y-z)/4=3/2-1/2-1/2=1/2.

证明完毕,收工.