max-value

[Mimmo]

How are you?

\(\displaystyle f(x,y)=xe^{-x-y}\)
What's the maximum value in the area encapsulated by the x- and y-axis and y = 1/2-x and y = 3/2 - x

\(\displaystyle \frac{\partial f}{\partial x} = e^{-x-y}(1-x)\)
\(\displaystyle \frac{\partial f}{\partial y} = -xe^{-x-y}\)

The first one is zero for x = 1 and the second for x = 0. Does this tell me no max/min exists??

Will I then have to search for the values on one of the lines y = 1/2-x or y = 3/2 - x?

Best regards!

 

[Gene]

I'm confused. Assuming you do mean
y=(1/2)-x and
y=(3/2)-x
the only purpose I can see for f(x,y) is
f₁(x,y)=f(x,(1/2)-x) and
f₂(x,y)=f(x,(3/2)-x) where you are trying for the area
(f₁-f₂)dx
But that gives an open triangle which doesn't touch either axis except at (0,0). No limit on the area.
I've been waiting for someone to understand the problem but...
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Gene