the only critical point is (1/3,1/3) in that region.
Just maximize
The two partial derivatives lead to
2x+y=1 and x+2y=1, which leads you to that one point in the first quadrant.
Then plug in and see that f(1/3,1/3)=1/27.
Let f(x,y)=xy(1-x-y).
Prove that max{f(x,y) | x>0 , y>0} = 1/27 .
I want to use the statement that sais that if f is continous in a compactic set A then f gets its minimum and maximum in A.
I'm not so sure on which compactic set I need to activate this statement...
Hope you'll be able to help me
Thanks in advance
I agree that we can tell that the partial deriatives are zero at this point...And that we can find that f(1/3,1/3)=1/27 ... But how can we prove that this point is the absolute maximum in the first quadrant ? I mean... How can we prove that there is no point such as in the first quadrant...??
I hope my question isn't too vague... Hope you'll be able to help me
Thanks a lot!