$f(x,y) = x^4 + y^4 -4xy +2$
$D = \{(x,y) | 0 \leq x \leq 3, 0 \leq y \leq 2\}$
I know how to find the absolute maximum and minimums of a function that is not bounded on a set D. However, I am not sure how take into account the boundaries in this problem.
Thanks
Just as in single variable calculus, you should locate all of the critical points (inside D) and then account for the "endpoints" as well. In this problem your "endpoints" are the edges: $x=0$, $x=3$, $y=0$, and $y=2$.
Usually to maximize or minimize along an edge you would use the method of Lagrange multipliers. However, for this problem it would be easiest to take each edge, "plug it in", and deal with it.
For example, to deal with $x=0$: $f(0,y)=0^4+y^4-4(0)y+2= y^4+2$. I want to find the min/max of $f(0,y)$ for $0 \leq y \leq 2$. It's only critical point is at $y=0$. So I need to test $y=0$ along with the endpoints ($y=0$ and $y=2)$. This gives me $f(0,0)=2$ and $f(0,2)=2^4+2=18$. So along this edge the min/max values are $2$ and $18$.
Now deal with the other three edges and any critical points inside $D$ and you'll find your absolute min/max values.