#### Answer

$z(1,2) =17$, $z(2,10) =70$, $z(7,5) =65$, $z(8,3) =58$.
maximum $z(2,10)=70$
minimum $z(1,2)=17$

#### Work Step by Step

Step 1. Given the objective function $z(x,y)=5x+6y$, we can obtain the function values at each corner as
$z(1,2)=5(1)+6(2)=17$, $z(2,10)=5(2)+6(10)=70$,
$z(7,5)=5(7)+6(5)=65$, $z(8,3)=5(8)+6(3)=58$.
Step 2. We can find the maximum of $z$ as $z(2,10)=70$,
Step 3. We can find the minimum of $z$ as $z(1,2)=17$,