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Nature of multivariate points 
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#1
Jan2012, 11:21 AM

P: 442

1. The problem statement, all variables and given/known data
Hi, I have been given a set of functions for which I need to find the stationary points , and determine whether the points are saddle, or max/min. I think I may have solved it correctly but I end up with all the points being saddle, surely this can't be right.. I may have gone wrong with my arithmetic. Can anyone go through my working. Appreciate the replies. 2. Relevant equations 3. The attempt at a solution [itex] f(x,y) = x^4 +(2x^2)y 4x^2 +3y^2 [/itex] [itex]F_{x}[/itex] = [itex]4x^3 +4xy8x [/itex] [itex]F_{y}[/itex] =[itex]2x^2+6y [/itex] [itex]F_{xx}[/itex]12x^2 +4y 8 [itex]F_{yy}[/itex]6 [itex]F_{yx}[/itex]4x The definition which I have used for delta/ determinant is : If Δ > 0 then stationary points are saddle i.e [itex]f_{xy}^2[/itex]  [itex]f_{xx}[/itex] * [itex]f_{yy}[/itex] The points which I get are the following: Re arranging (eq.1) and using eq. 2 (2x^2 = 6y) 4x^3 +4xy8x (eq.1) => x(4x^2 +4y8) = 0 x[ (2x^2 +2x^2 ]+4y8 = 0 x[ (6y6y) +4y 8] =0 x(12+4y8) = 0 x = 0 , 8y = 8 , y=1 Points are: (0,0) , (+/√3, 1) Thanks! 


#2
Jan2012, 12:12 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682

The point is that [itex]f_{xy}^2 f_{xx}f_{yy}[/itex] is the determinant of the "second derivative matrix" [tex]\begin{bmatrix}f_{xx}& f_{xy} \\ f_{xy} & f_{yy}\end{bmatrix}[/tex] of course that determinant is independent of the coordinate system and since this is a symmetric matrix, there exist a coordinate system in which it is diagonal. That is there exist a coordinate system in which the matrix is [tex]\begin{bmatrix}f_{xx} & 0 \\ 0 & f_{yy}\end{bmatrix}[/tex] which means that, locally, it is like [itex]f_{xx}x^2+ f_{yy}y^2[/itex]. If the determinant is negative, one of those coefficients is positive, the other negative, a saddle point. If the determinant is positive, the coefficients are either both positive, a local minimum, or both negative, a local maximum. 


#3
Jan2012, 02:40 PM

P: 442

You see on my lecturer's note I can clearly see that he has stated that when a coordinate system has Δ >0 then it's a saddle point , he did mention that " some books use the expression the other way around" , which I have confirmed upon browsing. Could he be wrong? Also thanks for your indepth answer ( as always). 


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