2. [-/33.33 Points] DETAILS
Find the exact extreme values of the function
$z = f(x, y) = (x - 4)^2 + (y - 1)^2 + 42$
subject to the following constraints:
$0 \le x \le 20$
$0 \le y \le 12$
Start by listing all nine candidates, including their z values, in the form (x, y, z):
First, list the four corner points and order your answers from smallest to largest x, then from smallest to largest y.
1) (
2) (
3) (
4) (
Next find the critical point.
5) (
Lastly, find the four boundary points and order your answers from smallest to largest x, then from smallest to largest y
6) (
7) (
8) (
9) (
Finally, find the extreme values:
$f_{min} = \text{____ at } (x, y) = (\text{____, ____})$
$f_{max} = \text{____ at } (x, y) = (\text{____, ____})$
Note that since this is a closed and bounded feasibility region, we are guaranteed both an absolute maximum and absolute minimum value of the function on the region.
