When I try and solve for critical points i end up with x=y. Where the domain for the function is inclusivly between $\displaystyle 0$ and$\displaystyle 2\pi$.

If I have infinitely many critical points is there another method to solving this?

$\displaystyle f(x,y)=sin(x)+sin(y)+sin(x+y)$

$\displaystyle f_x(x,y)=cos(x) + cos(x+y)$

$\displaystyle f_y(x,y)=cos(y) + cos(x+y)$

Now $\displaystyle f_x=0=f_y$

So when i set them equal-->

$\displaystyle cos(x) + cos(x+y)=cos(y) + cos(x+y)$

$\displaystyle cos(x)=cos(y)$

$\displaystyle x=y$

Now I am unsure what to do next, normally i get some critical points, but instead i got infinately many critical points...