Second derivative test for critical points

[pleasehelp.dontunderstand]

Hello! I need help with some information I am having difficulty finding.

I understand how to find critical points and I understand that when you plug the critical points into Fxx * Fyy - (Fxy)^2 that if the result is zero, then the test is inconclusive and if the result is negative, then the critical point is a saddle point. I also understand that if the result is positive, then you look back at Fxx and plug in the critical points and if the result is negative, then the critical point is a relative max and if the result is positive, then the critical point is a relative min. But what if Fxx gives you zero? What information does this yield?

Thank you very much!

 

[Steven G]

If F_xx=0 then what will F_xx * F_yy - (F_xy)^2 equal?

 

[pleasehelp.dontunderstand]

Okay, I found the mistake I made in the homework problem I was attempting (squared a negative incorrectly ?‍).

So is this scenario just not possible? I think I understand why - because if Fxx is zero, then Fxx * Fyy will be zero and you then must subtract something that is being squared (which must be either positive or zero) thus producing either a negative number (saddle point) or zero as the final result.