Question

f(x,y)=kx^2 2kxy y^2,在(0,0)处取极小值,求k的取值范围

f(x,y)=kx^2 2kxy y^2,在(0,0)处取极小值,求k的取值范围

Answer 1

z=f(x,y)=kx²+2kxy+y²
∂z/∂x=2kx+2ky
∂z/∂y=2y+2kx
∂²z/∂²x=2k
∂²z/∂²y=2
∂²z/∂x∂y=2k
驻点(0,0)处取极小值
∴2k>0→k>0
4k²-4k<0 0<k<1
∴k的取值范围0<k<1