设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2

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设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2作差法(x2+y2)

设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2
设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2

设x,y∈R,比较(x*2+y*2)*2与xy(x+y)*2
作差法
(x2+y2)2-xy(x+y)2
=x^4+y^4+2x^2y^2-x^3y-2x²y²-xy^3
=(x-y)(x^3-y^3)
=(x-y)²[(x+y/2)^2+3y²/4]
x=y则两式相等
x≠y则
(x-y)²>0,(x+y/2)²+3y²2/4>0
∴(x²+y²)²>xy(x+y)²