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?

The Hessian matrix of $h$ at $(1,0)$ is $\left[\begin{smallmatrix}2&2a\\2a&2\end{smallmatrix}\right]$, whose trace is $4$ and whose determinant is $4-4a^2$. If $\lvert a\rvert>1$, then $h$ will have then a saddle point at $(1,0)$. If $\lvert a\rvert<1$, $h$ will have a minimum at $(1,0)$. And if $a=\pm1$, then$$h(x,y)=\bigl((x-1)\pm y\bigr)^2+y^4.$$Can you take it from here?
