critical points

[SigepBrandon]

If Fxx(a,b)=0 does that mean that (a,b) is not a critical point?

 

[Deleted member 4993]

If Fxx(a,b)=0 does that mean that (a,b) is not a critical point?

For a function of multiple variable, you need to consider the determinant of the Hessian Matrix.

 

[SigepBrandon]

Thanks Sub!
I don't think we've got to that point yet. We've instead been given the formula for the discriminant and a couple cases to identify max/min and saddle points. Those conditions are as follows:

D=F[sub:39t0q10u]xx[/sub:39t0q10u](a,b)F[sub:39t0q10u]yy[/sub:39t0q10u](a,b)-[F[sub:39t0q10u]xy[/sub:39t0q10u](a,b)][sup:39t0q10u]2[/sup:39t0q10u]

1] If D>0 and F[sub:39t0q10u]xx[/sub:39t0q10u]>0, then F(a,b) is a local min for f
2] If D>0 and F[sub:39t0q10u]xx[/sub:39t0q10u]<0, then f(a,b) is a local max for f
3] If D<0, then f has a saddle point at (a,b)
My notes continue on to say that the test fails for D=0 and points near (a,b) need to be considered to discover the classification.

That's about all I've been given so far, I'm just not sure how that applies if F[sub:39t0q10u]xx[/sub:39t0q10u]=0.

 

[SigepBrandon]

In the exercises I've worked, the discriminant when F[sub:2tpju1wy]xx[/sub:2tpju1wy](a,b)=0 is less than 0, so I'm assuming that regardless of what F[sub:2tpju1wy]xx[/sub:2tpju1wy](a,b) is, when D<0 it's always a saddle point? maybe this is pretty obvious to most, it wasn't, isn't too me. For some reason I find the above "rules" ambiguous.