how to find the nature of a critical point

Question

I have this function $$ f(x,y)=-2(x-y)^2 +x^4+y^4 $$
I need to find the nature of the critical point $(0,0)$ (we can't conclude using the Hessian Matrix) so I notice that all critical points verify the equation $y=-x$ and then calculate $f(x,-x)$ $$ f(x,-x) = 2x^4-8x^2$$ and then use Taylor's theorem to get $$ f(x,-x) \approx -8x^2 $$ so the point $(0,0)$ is a local maximum but as I check the solution it states that it's a saddle point. Can anyone check where I messed up? sorry if there's a bad vocabulary usage as I'm studying math in French

Answer 1

Don't see only those paths where other critical points lie. It is a saddle point, since if you go along $y=-x$ (as you stated), it is a local maximum. But if you go along $y=x$, then $$f(x,x)=2x^4$$ Therefore, origin is the point of local minimum here.