Question

f(x,y)=(y-x^2)(y-x^4),求极值

Answer 1

令f(x,y)=(y-x^2)(y-x^4)=z
则z'x=dz/dx=(y-x^2)‘(y-x^4)+(y-x^2)(y-x^4)’=-2x(y-x^4)+(y-x^2)(-4x^3)
z'y=dz/dy=(y-x^2)‘(y-x^4)+(y-x^2)(y-x^4)’=(y-x^4)+(y-x^2)=2y-x^4-x^2
令z'x=0 z'y=0
则有
1.x=y=0
2.y=3/8 x^2=1或者x^2=1/2
驻点为（0,0）（±1,3/8）（±0.5√2,3/8）
显然极小值为z极小=f(x,y)=f(±0.5√2,3/8)=-1/64
z极大=f(x,y)=f(±1,3/8)=0