Simple critical point question - how to specify?

[Gekko]

Homework Statement

Classify the critical points subject to the constraints:
0<=x, y<=Pi

Homework Equations

fx=sin(y)sin(2x+y)
fy=sin(x)sin(2y+x)

The Attempt at a Solution

Clearly one set of critical points will be (n*Pi,m*Pi) where m <=1 and n and m are all positive and negative integers (Z)
Another will be where 2x=y and where 2y = x but this is only true where x and y = 0
Another is where 2x+y = n*Pi and 2y + x = m*Pi

I think that is it covered however how do I write this in a precise mathematical way?

 

[HallsofIvy]

Homework Statement

Classify the critical points subject to the constraints:
0<=x, y<=Pi

Homework Equations

fx=sin(y)sin(2x+y)
fy=sin(x)sin(2y+x)

The Attempt at a Solution

Clearly one set of critical points will be (n*Pi,m*Pi) where m <=1 and n and m are all positive and negative integers (Z)
Another will be where 2x=y and where 2y = x but this is only true where x and y = 0

Great!

Another is where 2x+y = n*Pi and 2y + x = m*Pi

Solve this pair of equations for x and y. The solutions, of course, will depend on "mPi" and "nPi" and I suspect (0, 0) will be included in it.

I think that is it covered however how do I write this in a precise mathematical way?

 

[Gekko]

Thanks a lot for your replies. So is it enough to simply say:

y=1/3 Pi(2m-n) where 2m-n<3

x=1/2 Pi(2n-m) where 2n-m=>0 or 2n=>m

Is this enough to describe the critical points in a mathematically correct way?