Maxima and Minima for a two variable function

[Odyssey]

Greetings, can you guys please help me with my assignment??

I am supposed to find and classify all critical points of the function f(x,y) = (sin x)(cos y)

Now I took the first partials with repsect to x and y and they are (cos x)(cos y) = 0 and (-sin x)(sin y) = 0, respectively.

Now I know fx = 0 when either cos x = 0 or cos y = 0 and cos x = 0 when x is pi/2, -pi/2, 3pi/2, -3pi/2, ...

and fy = 0 when either sin x = 0 or sin y = 0 and sin x = 0 when x = kpi, k being integers...

But I don't know if there are infinitely many critical points or no critical points?? If there are infinitely many, where are they at??

Thanks for the help!

 

[whozum]

There are critical points as you showed, and there are infinitely many of them as you know from a sin(x) or cos(x) graph.