Q:

Find absolute maximum and minimum of a function with two variables

I have the function$ f(x,y)=(x+y-2)^2$
I have the constraints $0\leq x \leq 3$ and $x \leq y \leq 3$
The partial derivatives are $f_x =2(x+y-2)$ and $f_y =2(x+y-2)$
So the stationary points in the domain are (0,2) and (1,1)
Now if I look at the second partial derivative it isn't helpful at $f_{xx} =2 f_{yy}=2$ and $f_{xy} =2$
How can I find the absolute maximum and minimum thanks

A:

You'll have that all of the points $(x,y)$ with $x+y = 2$ will be critical points, not only $(1,1)$ and $(0,2)$. The function is zero at these points and $f\geq 0 $ always, so each one of them is an absolute minimum. 
You can also look at $(x+y-2)^2$ with $x+y$ as a variable on its own. Given that $0\leq x\leq y \leq 3$, its greater value when $x=y=3$, so $(3, 3)$ is the maximum point with $f(3,3) = 16 $.

