How to infer critical points of functions containing periodic functions such as trigonometric functions?
E.g. $f: \mathbb{R}^2 \rightarrow \mathbb{R}$
$$f(x,y)=x+y \sin(x)$$
One can discover that there seem to be infinite amount of critical points, since critical point is attained whenever one reaches a particular $\sin(x)$ and $\cos(x)$ value and these continue to $- \infty$ and $\infty$.
So how does one then infer about the critical points? Or can one limit oneself to one interval since the others will behave exactly the same (i.e. the function and its gradient are symmetric?)?
The critical points are easy to compute: we have
$f_x(x,y)=1+y \cos(x)$ and $f_y(x,y)=\sin(x)$
Hence $f_y(x,y)=0$ iff $x= k \pi$ for some $k \in \mathbb Z$. Then
$f_x( k \pi,y)=1+y(-1)^k=0$ iff $y=(-1)^{k+1}$
Thus the critical points are
$$( k \pi, (-1)^{k+1}),$$
$k \in \mathbb Z$.