求函数f(x,y)=xy(2-x-y)的极值

网友回答

f'x=y(2-x-y)-xy=y(2-2x-y)=0,得y=0或y=2-2x
f'y=x(2-x-y)-xy=x(2-x-2y)=0,得x=0或x=2-2y
解得有以下几组(0,0), (2,0), (0, 2), (2/3,2/3)
A=fxx=-2y
B=fxy=2-2x-2y
C=fyy=-2x
在(0,0), AC-B^2=0-2^2=-4在(2,0),A=0, B=-2, C=-4, AC-B^2=-4在(0,2),A=-4, B=-2, C=0, AC-B^2=-4在(2/3,2/3), A=-4/3, B=-2/3,C=-4/3, AC-B^2=16/9-4/9=4/3>0, 且A