已知x,y,z∈(0,+∞)求证:√x^+xy+y^+√y^+yz+z^+√z^+zx+x^>=3/2(x+y+z )
问题描述:
已知x,y,z∈(0,+∞)求证:√x^+xy+y^+√y^+yz+z^+√z^+zx+x^>=3/2(x+y+z )
答
√(x^2+xy+y^2) = √[(x^2+xy+1/4y^2)+3/4y^2]>√(x^2+xy+1/4y^2)=x^2+1/2y^2同理√y^2+yz+z^2>y^2+1/2z^2 √z^2+zx+x^2>z^2+1/2z^2相加得√x^2+xy+y^2+√y^2+yz+z^2+√z^2+zx+x^2>=3/2(x+y+z )