I have the expression:
and have to find the critical points and their nature. The points found are:
$(1,1) = $ saddle point
$(1,-1) = $ saddle point
$(\frac 3 4 ,0) = $ global minimum
$(0,0) = $ inconclusive, however, when I take $0.1$ and $-0.1$ as $2$ points slightly higher or lower to find the nature of the point, I get a value positive or negative value for both the $fxx$ and the Hessian Determinant i.e. $D < 0$ at $0.1$ and $D > 0$ at $-0.1$.
I understand, for a single variable function, that two different signs for values higher or lower than the points, would mean the point $(0,0)$ is an inflection point. But since it is a multi-variable function, how can I determine the nature of $(0,0)$ since the Hessian Determinant must be $< 0 $ for the points to be saddle points?