x/(1+x)+y/(1+y)+z/(1+z)=1 求证x+y+z大于等于2(xy+yz+zx)

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/19 07:49:02
x/(1+x)+y/(1+y)+z/(1+z)=1求证x+y+z大于等于2(xy+yz+zx)x/(1+x)+y/(1+y)+z/(1+z)=1求证x+y+z大于等于2(xy+yz+zx)x/(1+x

x/(1+x)+y/(1+y)+z/(1+z)=1 求证x+y+z大于等于2(xy+yz+zx)
x/(1+x)+y/(1+y)+z/(1+z)=1 求证x+y+z大于等于2(xy+yz+zx)

x/(1+x)+y/(1+y)+z/(1+z)=1 求证x+y+z大于等于2(xy+yz+zx)
∵x/(1+x)+y/(1+y)+z/(1+z)=1
  ∴(x(x+1)+y(y+1)+z(z+1))(x/(1+x)+y/(1+y)+z/(1+z))≥(x+y+z)²
  即x²+x+y²+y+z²+z≥(x+y+z)²
  将右边展开即x+y+z≥2(xy+yz+zx)