Question

a staionary point of f(x,y)=x2-xy+y2-2x+y is

Answer 1

Answer:

∂f∂x=y−2x−y2

∂f∂y=x−2xy

The second one vanishes for x=0 or y=1/2. Substituting in the first one we get

(0,0),(0,1),(1/8,1/2)

Now consider the Hessian matrix

Hf(x,y)=⎡⎣⎢∂2f∂x2∂2f∂y∂x∂2f∂x∂y∂2f∂y2⎤⎦⎥=[−21−2y1−2y−2x]

Now compute the Hessian at each of the three point and look whether the associated bilinear form is positive definite, negative definite or indefinite.