I just have question about which method I should be using to do the second derivative test: For the function: $f(x,y)=x^3y+xy^3-xy+1$ at the point (0,0).
I did this by solving: det$[H_{f}(0,0) - \lambda I] = 0$
Calculating the partial derivatives and substuting in $(0,0)$, I get det$\begin{smallmatrix} -\lambda & -1\\ -1 & -\lambda\\ \end{smallmatrix} = \lambda^2 - 1$, so $\lambda = \pm 1$ - so $(0,0)$ is Saddle point.
Is this a valid way of doing this? Or should I simply use that det $H_{f}(0,0)$ = $\begin{smallmatrix} 0 & -1\\ -1 & 0\\ \end{smallmatrix} = -1 < 0$, which also indicates that it is a saddle point.
Are both methods valid for function of 2 variables?
