f(x,y)=15x^2+3y^2+1-(3x^2+y^2+1)^2在x^2+y^2
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f(x,y)=15x^2+3y^2+1-(3x^2+y^2+1)^2在x^2+y^2
![f(x,y)=15x^2+3y^2+1-(3x^2+y^2+1)^2在x^2+y^2](/uploads/image/z/19966068-36-8.jpg?t=f%28x%2Cy%29%3D15x%5E2%2B3y%5E2%2B1-%283x%5E2%2By%5E2%2B1%29%5E2%E5%9C%A8x%5E2%2By%5E2)
因为:x²+y²≤1,
所以:y²≤1-x²……………………(1)
且:|x|≤1,|y|≤1…………………(2)
f(x,y)=15x²+3y²+1-(3x²+y²+1)²
f(x,y)=15x²+3y²+1-9x^4-y^4-1-6x²y²-6x²-2y²
f(x,y)=-(9x^4+6x²y²+y^4)+3(3x²+y²)
f(x,y)=-(3x²+y²)²+3(3x²+y²)
f(x,y)=-(3x²+y²)(3x²+y²-3)
-f(x,y)=(3x²+y²)(3x²+y²-3)
将(1)代入,有:
-f(x,y)≤(3x²+1-x²)(3x²+1-x²-3)
-f(x,y)≤(2x²+1)(2x²-2)
将(2)代入,有:
-f(x,y)≤(2×1²+1)(2×1²-2)
-f(x,y)≤3×0
-f(x,y)≤0
f(x,y)≥0
所以:y²≤1-x²……………………(1)
且:|x|≤1,|y|≤1…………………(2)
f(x,y)=15x²+3y²+1-(3x²+y²+1)²
f(x,y)=15x²+3y²+1-9x^4-y^4-1-6x²y²-6x²-2y²
f(x,y)=-(9x^4+6x²y²+y^4)+3(3x²+y²)
f(x,y)=-(3x²+y²)²+3(3x²+y²)
f(x,y)=-(3x²+y²)(3x²+y²-3)
-f(x,y)=(3x²+y²)(3x²+y²-3)
将(1)代入,有:
-f(x,y)≤(3x²+1-x²)(3x²+1-x²-3)
-f(x,y)≤(2x²+1)(2x²-2)
将(2)代入,有:
-f(x,y)≤(2×1²+1)(2×1²-2)
-f(x,y)≤3×0
-f(x,y)≤0
f(x,y)≥0
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