Finding the critical points is not easy.
I don't think it can be solved analytically(I could be wrong)
I drew a graph to see what it looked like and here it is.
So we want to solve the system of equations
subbing the first into the 2nd we get
finally we get
If we sub this back into the first equation we get
Finally we can use newtons method on the last equation
Based on the graph from above I used
After applying newtons method three times I got the values(on a calculator)
Oddly enough when I repeated the process but eliminted the x's and used newtons method again I got the same value for y.
so the critical numbers are )
I hope this helps.
P.S if anyone knows another way I would love to see it
Thanks, The Empty Set.
I have been thinking about this and just noticed that
and
Are inverses of each other. So they are symmetric over the line y=x
This means that any of their intersections would have to be on the line y=x.
So if I sub this into the first equation I get
So the exact value is