求z=e^(X-Y)(X^2-2Y^2)的极值

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求z=e^(X-Y)(X^2-2Y^2)的极值求z=e^(X-Y)(X^2-2Y^2)的极值求z=e^(X-Y)(X^2-2Y^2)的极值Fx(x,y)=e^(x-y)(x^2-2y^2)+e^(x-

求z=e^(X-Y)(X^2-2Y^2)的极值
求z=e^(X-Y)(X^2-2Y^2)的极值

求z=e^(X-Y)(X^2-2Y^2)的极值
Fx(x,y)= e^(x-y)(x^2-2y^2)+e^(x-y)·2y=e^(x-y)·(x^2+2x-2y^2)=0 ①
Fy(x,y)=- e^(x-y)(x^2-2y^2)+e^(x-y)·(-4y)=e^(x-y)·(x^2+4y-2y^2)=0 ②
由 ①②得驻点为 (0,0) (-4,-2)
A= Fxx(x,y)= e^(x-y)(x^2+2x-2y^2)+e^(x-y)·(2x+2)=e^(x-y)·(x^2+4x-2y^2+2)
B= Fxy(x,y)= -e^(x-y)(x^2+2x-2y^2)+e^(x-y)·(-4y)=-e^(x-y)·(x^2+2x-2y^2+4y)
C= Fyy(x,y)= e^(x-y)(x^2-2y^2+4y)-e^(x-y)·(-4y+4)=e^(x-y)·(x^2-2y^2+8y-4)
在点(0,0)处 AC-B^2 ＜0 舍去
在点(-4,-2)处 AC-B^2＞0 A＜0
则z在 (-4,-2)处有极大值 f（-4,-2）= 8e^(-2)

在（0,2）处取极大值为-8e^2

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