Question

relative extreme points of sin(x)+sin(y)

Answer 1

Call
\[
f(x,y)=\sin(x)+\cos(y)\\
\nabla f(x,y)=(\cos(x),\sin(y))\\
\]
The first step is to find when the gradient is (0,0).
Try to do it.

Answer 2

You can see from the plot that there are infinite numbers of relative max and min
http://www.wolframalpha.com/input/?i=plot3d+sin%28x%29%2Bcos%28y%29+++from+-4pi+to+4pi

Answer 3

In my question for both parts (after derivative ) i.e cos(x) and cos(y) both =0 at 90 and 270 . so x= 90,270 and y=90,270 .now if i have to calculate the value of function . what value of x and y i sholud put in main function i.e sin(x) =sin(y) ??

Answer 4

You are correct