实数x、y满足x^2+xy+y^2=1,则F(x,y)=x^3y+xy^3的最大值与最小值之和为?
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![实数x、y满足x^2+xy+y^2=1,则F(x,y)=x^3y+xy^3的最大值与最小值之和为?](/uploads/image/z/14575052-20-2.jpg?t=%E5%AE%9E%E6%95%B0x%E3%80%81y%E6%BB%A1%E8%B6%B3x%5E2%2Bxy%2By%5E2%3D1%2C%E5%88%99F%28x%2Cy%29%3Dx%5E3y%2Bxy%5E3%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E4%B8%8E%E6%9C%80%E5%B0%8F%E5%80%BC%E4%B9%8B%E5%92%8C%E4%B8%BA%3F)
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实数x、y满足x^2+xy+y^2=1,则F(x,y)=x^3y+xy^3的最大值与最小值之和为?
实数x、y满足x^2+xy+y^2=1,则F(x,y)=x^3y+xy^3的最大值与最小值之和为?
实数x、y满足x^2+xy+y^2=1,则F(x,y)=x^3y+xy^3的最大值与最小值之和为?
G(x,y)=x^3y+xy^3+a(x^2+xy+y^2-1)
Gx`(x,y)=3x^2y+y^3+a(2x+y)=0
Gy`(x,y)=x^3+3xY^2+a(x+2y)=0
3x^2y+y^3+a(2x+y)-{x^3+3xY^2+a(x+2y)}=0
(x-y)(-x^2-y^2+2xy+a)=0
1)x=y
x^2+xy+y^2=1
3x^2=1
x=±1/√3
y=±1/√3
F(x,y)=x^3y+xy^3=2/9
x^3y+xy^3
=xy(x^2+y^2)
=xy(1-xy)
=-(xy-1/2)^2+1/4
又xy=1-(x^2+y^2)≤1-2xy
∴xy≤1/3
x^2+xy+y^2=1=(x+y)^2-xy≥-xy
∴xy≥-1当x=1,y=-1或y=1,x=-1可取到等号
∴最大值与最小值之和为
-(-1-1/2)^2+1/4-(1/3-1/2)^2+1/4=-16/9
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