if x and y don't equal zero, then surely you can divide through by y in equation 1, x in equation 2. Then you have two things equal to 0... if you equate them what do you get?
This is just a starting step...
Hi!
I Can't seem to find the critical points of the function f(x,y) = x*y*ln(x^2 + y^2)
I did the df/dx = y*[ (2 x^2)/(x^2 + y^2) + ln(x^2 + y^2) ]
and the df/dy = x*[ (2 y^2)/(x^2 + y^2) + ln(x^2 + y^2) ]
I equalled them both to 0, but now I'm stuck because I don't know how to solve the system:
y*[ (2 x^2)/(x^2 + y^2) + ln(x^2 + y^2) ] = 0
x*[ (2 y^2)/(x^2 + y^2) + ln(x^2 + y^2) ] = 0
I have the solutions. There are 4 critical points and none of them is x=0 or y=0.
But I don't know how to get to the solution.
Thanks in advance!
Tony
if x and y don't equal zero, then surely you can divide through by y in equation 1, x in equation 2. Then you have two things equal to 0... if you equate them what do you get?
This is just a starting step...
I tried that but it got me nowhere. The solutions are:
[ 1/(sqrt(2)*e); 1/(sqrt(2)*e) ], [ -1/(sqrt(2)*e); -1/(sqrt(2)*e) ],
[-sqrt(e)/2 ; sqrt(e)/2] and [ sqrt(e)/2 ; -sqrt(e)/2 ].
I don't see how I could get this from (2 x^2)/(x^2 + y^2) + ln(x^2 + y^2) = 0
and (2 y^2)/(x^2 + y^2) + ln(x^2 + y^2) = 0
I tried adding them, substracting, moving the 'ln' to the other side and raising them as power of 'e', but I can't seem to solve it.
your last 2 critical points are wrong i'm afraid, I have spent the last 10 minutes puzzling over those solutions, but now I have graphed it, and I have attained the correct solution.
The set of solutions is...
I will post working shortly