$[12\mathrm{.}73/33\mathrm{.}33$ Points$]$

PREVIUUS AIVSVVERS

Find the exact extreme values of the function

$z=f(x,y)=(x-5{)}^2+(y-3{)}^2+22$

subject to the following constraints:

$0x18$

$0y12$

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 $y.$

$00()$

$(0$

$x,36$

$$

$x)$

$57)$

$([5x,0x],[4)],[5,6x,21x]$

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 $y.$

$,0x,$ x $)$

$(0x,6x,3Ex)$

$(5,0,0x,57x)$

$x,6$

$x,2121)$

Finally, find the extreme values:

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.

f

$$