高2不等式证明.设x.y属于0到正无穷证明1/4（x+y)+1/2(x+y)*2大于等于x被根号y+y倍根号x

证明：左边=[(x+y)/4]+[(x+y)²/2]=[(x+y)/4]×[1+2(x+y)].(一）∵x,y＞0.∴由基本不等式可得：x+y≥2√(xy).===>(x+y)/4≥(1/2)√(xy).①等号仅当x=y＞0时取得。（二）由基本不等式可得2x+(1/2)≥2√x.且2y+(1/2)≥2√y.两式相加可得:1+2(x+y)≥2(√x+√y).②.（三）将①，②两个不等式相乘可得[(x+y)/4]×[1+2(x+y)]≥(√x+√y)√(xy)=x√y+y√x.即[(x+y)/4]+[(x+y)²/2]≥x√y+y√x.

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