(adsbygoogle = window.adsbygoogle || []).push({}); 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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# Nature of multivariate points

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