求函数f(x,y)=e^x(x+2y+y^2)的极值

Question

求函数f(x,y)=e^x(x+2y+y^2)的极值

Answer 1

f'x(x,y)=e^x(x+2y+y^2+1)=0
f'y(x,y)=2e^x(1+y)=0
解得x=0 y=-1
A=f''xx(x,y)=e^x(x+2y+y^2+2)=1
B=f''xy(x,y)=2e^x(1+y)=0
C=f''yy(x,y)=2e^x=2
AC-B^2=2>0,A>0
所以函数在(0,-1)处取得极小值=-1