for what a does h have an extreme point in (1,0)? I got this function:
$h(x, y) = (x − 1)^2 + 2a(x − 1)y + y^2 + y^4$
For what $a$ does h have either a maxima- or a minima-point but not a saddle point in (1,0)?
I have confirmed that the point is stationary, but it gets really tricky when trying to use the Quadratic form when having $a$ around. Do you have any ideas on how to solve this?

HINT
We have that


*

*$h_x=2(x-1)+2ay\implies h_x(1,0)=0$

*$h_y=2a(x-1)+2y+4y^3\implies h_y(1,0)=0$
then $(1,0)$ is a stationary point as you have noticed.
Then we need to consider


*

*$h_{xx}=2\implies h_x(1,0)=2$

*$h_{yy}=2+12y^2\implies h_y(1,0)=2$

*$h_{xy}=h_{yx}=2a\implies h_y(1,0)=2a$
finally proceed by the hessian matrix test.
