已知x、y、z∈R+,求f(x,y,z)=(xy+2yz+2zx)/(x^2+y^2+z^2)最大值要速度的说

Question

已知x、y、z∈R+,求f(x,y,z)=(xy+2yz+2zx)/(x^2+y^2+z^2)最大值
要速度的说

Answer 1

若f(x,y,z)=(2xy+2yz+2zx)/(x^2+y^2+z^2)
∵2xy≤x²+y²,2yz≤y²+z²,2zx≤z²+x²,
∴f(x,y,z)=(2xy+2yz+2zx)/(x²+y+z²)≤2(x²+y²+z²)/(x²+y²+z²)=2,
当且仅当x=y=z时,等号成立,f(x,y,z)的最大值为2.