$z = f(x,y) = 3x^2 - 2xy + 4y^2 - 6x - 20y + 9$
subject to the following constraints:
$0 \le x \le 5$
$0 \le y \le 6$
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} = $ at $(x,y) = ($ , , )
$f_{max} = $ at $(x,y) = ($ , , )
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.
