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Nature of multivariate points

by ibysaiyan
Tags: multivariate, nature, points
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ibysaiyan
#1
Jan20-12, 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 +4xy-8x [/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 +4xy-8x (eq.1)
=>
x(4x^2 +4y-8) = 0
x[ (2x^2 +2x^2 ]+4y-8 = 0
x[ (-6y-6y) +4y -8] =0
x(-12+4y-8) = 0

x = 0 , -8y = 8 , y=-1
Points are:
(0,0) , (+/√3, -1)

Thanks!
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HallsofIvy
#2
Jan20-12, 12:12 PM
Math
Emeritus
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P: 39,682
Quote Quote by ibysaiyan View Post
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 +4xy-8x [/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]
You have this exactly backwards- the stationary points are saddles if Δ< 0.
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.

The points which I get are the following:
Re arranging (eq.1) and using eq. 2 (2x^2 = -6y)
4x^3 +4xy-8x (eq.1)
=>
x(4x^2 +4y-8) = 0
x[ (2x^2 +2x^2 ]+4y-8 = 0
x[ (-6y-6y) +4y -8] =0
x(-12+4y-8) = 0

x = 0 , -8y = 8 , y=-1
Points are:
(0,0) , (+/√3, -1)

Thanks!
ibysaiyan
#3
Jan20-12, 02:40 PM
P: 442
Quote Quote by HallsofIvy View Post
You have this exactly backwards- the stationary points are saddles if Δ< 0.
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.
Hi there,
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 in-depth answer ( as always).


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