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 March 18th, 2009, 12:53 PM #1 Member   Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0 Quick Second Partial Derivative Test Question I understand that when the discriminant of a function defined as $D(x,y)=F_{xx}(x,y)F{y,y}(x,y)-F_{xy}(x,y)^2$ is less than zero at point (a,b) then the function has a saddle point at (a,b) and if the discriminant is greater than zero at (a,b) there exists a local extrema at point (a,b) (D=0 is inconclusive). Also if there is a local extrema then: $F_{xx}(a,b)>0$ there exists a local minimum at (a,b) $F_{xx}(a,b)<0$ there exists a local maximum at (a,b) However what about the case that D>0 at (a,b) indicating that there is a local extrema, but $F_{xx}(a,b)=0$ ? This can't be a saddle point can it? Or does the test fail?
 March 18th, 2009, 11:38 PM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Quick Second Partial Derivative Test Question In that case look at $F_{yy}(a,b)$ instead. Also: the singular is extremum, the plural is extrema.
 March 19th, 2009, 07:51 PM #3 Member   Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0 Re: Quick Second Partial Derivative Test Question Ahh ok thank you. Is there any particular reason we choose to evaluate $f_{xx}$ first rather than $f_{yy}$?
 March 19th, 2009, 09:21 PM #4 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Quick Second Partial Derivative Test Question No. You can check any directional derivative.

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