已知实数xy满足等式3x²+2y²-6x=0,求w=x²+y²的取值范围

已知实数xy满足等式3x²+2y²-6x=0,求w=x²+y²的取值范围
已知实数xy满足等式3x²+2y²-6x=0,求w=x²+y²的取值范围

已知实数xy满足等式3x²+2y²-6x=0,求w=x²+y²的取值范围
3x²+2y²-6x=0
y² = -3/2x²+3x = -3/2x(x-2) ≥ 0
0≤x≤2
w = x²+y² = x²-3/2x²+3x = -1/2x²+3x =-1/2(x-3)²+9/2
0≤x≤2
-3≤x-3≤1
1≤(x-3)²≤9
-9/2≤-1/2(x-3)²≤-1/2
0≤-1/2(x-3)²+9/2≤4
取值范围【0,4】

由3x^2+2y^2=6x得
2y^2=-3(x^2-2x+1)+3=-3[(x-1)^2-1]
由于y^2>=0
所以
-3[(x-1)^2-1]>=0
(x-1)^2<=1
0<=x<=2
又3x^2+2y^2=6x可转化为
2(x^2+y^2)=6x-x^2=-(x^2-6x+9)+9=-(x-3)^2+9
当x=2时,...

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由3x^2+2y^2=6x得
2y^2=-3(x^2-2x+1)+3=-3[(x-1)^2-1]
由于y^2>=0
所以
-3[(x-1)^2-1]>=0
(x-1)^2<=1
0<=x<=2
又3x^2+2y^2=6x可转化为
2(x^2+y^2)=6x-x^2=-(x^2-6x+9)+9=-(x-3)^2+9
当x=2时,-(x-3)^2+9有最大值8.
故当x=2时,
x^2+y^2有最大值4.
当X=0时,有最小值是:0
故w的范围是[0,4]

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