∵x²+y²+z²=[x²+(y²)/5]+[(4y²)/5+z²]≥2√5(xy)/5+2√5(2yz)/5=(2√5)(xy+2yz)/5
∴(xy+2yz)/(x²+y²+z²)≤5/(2√5)=√5/2
当且仅当 x=y/√5 2y/√5=z时等号成立
所以 (xy+2yz)/(x²+y²+z²)的最大值是√5/2