1/4(x+y)+1/2(x+y)^2
=(1/4)x+(1/4)y+(1/2)x^2+xy+(1/2)y^2
=((1/2)x^2+(1/2)y^2)+(1/4)x+(1/4)x+xy
因为x＞0,y＞0
所以x^2＞0,y^2＞0
所以,由正弦定理得
(1/2)x^2+(1/2)y^2≥xy
所以,原式≥2xy+(1/4)x+(1/4)y=((1/4)x+xy)+((1/4)y+xy)
因为x＞0,y＞0
所以(1/4)x+xy≥x√y同理可得(1/4)y+xy≥y√x
所以原式≥2xy+(1/4)x+(1/4)y=((1/4)x+xy)+((1/4)y+xy)≥x√y+y√x