f(x)=x²-ax,x>y>0,且xy=2,若f(x)+f(y)+2ay≥0恒成立,求a的取值范围
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![f(x)=x²-ax,x>y>0,且xy=2,若f(x)+f(y)+2ay≥0恒成立,求a的取值范围](/uploads/image/z/4032649-1-9.jpg?t=f%28x%29%3Dx%26%23178%3B-ax%2Cx%3Ey%3E0%2C%E4%B8%94xy%3D2%2C%E8%8B%A5f%28x%29%2Bf%28y%29%2B2ay%E2%89%A50%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
f(x)=x²-ax,x>y>0,且xy=2,若f(x)+f(y)+2ay≥0恒成立,求a的取值范围
f(x)=x²-ax,x>y>0,且xy=2,若f(x)+f(y)+2ay≥0恒成立,求a的取值范围
f(x)=x²-ax,x>y>0,且xy=2,若f(x)+f(y)+2ay≥0恒成立,求a的取值范围
x²-ax+y²+ay≥0
(x²+y²)/(x-y)≥a
[(x-y)²+2xy]/(x-y)≥a
(x-y)+[4/(x-y)]≥a
均值不等式得(x-y)+[4/(x-y)]≥4
于是a≤4
F(X)+F(Y)+2AY>=0等价于X^2+Y^2+A(Y-X)>=0 即A<=(X^2+Y^2)/(X-Y)=(X^2+Y^2-4)/(X-Y)+4/(X-Y)=X-Y+4/(X-Y)
当X-Y=2时X-Y+4/(X-Y)取得最小值4,所以A<=4