.
Then use calculus to find these values precisely.
Solution: Here is the graph on the specified domain. Apparently, there is one max, one saddle point, and one minimum all along the line y = x.
The contour confirms theis initial conjecture:
To continue on to the partials and the extreme points, we need to solve:
Using the conjecture that the critical points lie along the line y = x we set y = x in one of these partials to get cos(x)+cos(2x) = 0, which has the solutions x = Pi/3, x = 5Pi/3, and x = Pi.
Setting (x,y) = (Pi/3, Pi/3), (Pi, Pi), and (5Pi/3, 5Pi/3) in the two equations above, we see that these three points are extreme points. Furthermore, by the graphs, there are no other extreme points in the given domain.
Evaluating, there is a maximum of
at x = Pi/3,y = Pi/3, a saddle at (0,0) and a minimum of
-
at x = 5Pi/3 and y = 5Pi/3.
A graph of f(x,y) on the boundary shows that there are 9 local extrema on the boundary.
Since the extrem values at the interior points are
it follows that the golbal extrema occur at these interior points.