# Thread: 2 Variables Calculus Help

1. ## 2 Variables Calculus Help

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

2. the only critical point is (1/3,1/3) in that region.

Just maximize $\displaystyle f(x,y)=xy-x^2y-xy^2$

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.

3. 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 $\displaystyle f(x,y) > 1/27$ in the first quadrant...??

I hope my question isn't too vague... Hope you'll be able to help me

Thanks a lot!

4. First derivatives are zero, then
df=(1/2)(f_xx*dx^2+2*f_xy*dxdy+f_yy*dy^2)
f_xx=-2/3
f_yy=-2/3
f_xy=-1/3
df=-1/3(dx^2+dxdy+dy^2)
dx=rcos(a)
dy=rsin(a)
df=(-1/3)r^2(1+cos(a)sin(a))=(-1/3)r^2(1+sin(2a)/2)<0
this point is absolute maximum.