求函数f(x,y)=x³-y³+3x²+3y²-9x的极值.
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求函数f(x,y)=x³-y³+3x²+3y²-9x的极值.
求函数f(x,y)=x³-y³+3x²+3y²-9x的极值.
求函数f(x,y)=x³-y³+3x²+3y²-9x的极值.
∂f/∂x=3x^2+6x-9=0 x²+2x-3=(x-1)(x+3)=0 x1=1,x2=-3
∂f/∂y=-3y²+6y=0 y²-2y=y(y-2) y1=0 y2=2
A=∂²f/∂x²=6x+6
C=∂²f/∂y²=-6y+6
B=∂²f/∂x∂y=0
△=AC-B²
得到f(x,y)的四个稳定点:P1(1,0),P2(1,2),P3(-3,0),P4(-3,2)
(1)对P1点:A=12>0,B=0,C=6>0 △=AC-B²=72>0 f(1,0)=-5 为极小值;
(2)对P2,P3点不是极值点;
(3)对P4点:A=-12