The feasible solution for a $LPP$ is shown in the figure. Let $z=3x-4y$ be the objective function. The minimum value of $z$ occurs at:

The corner points of the bounded feasible region are $(60,0), (120,0), (60,40), (40,20)$ and $(20,30)$. For the objective function $z=5x+10y$:
$(i)$ Maximum value of $z$.
$(ii)$ Minimum value of $z$.
$(iii)$ Maximum value of $z$ occurs at.
$(iv)$ Minimum value of $z$ occurs at.

The optimal value of the objective function is attained at the points

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

For the $LP$ problem,minimize $z = 2x + 3y$,the coordinates of the corner points of the bounded feasible region are $A(3, 3), B(20, 3), C(20, 10), D(18, 12),$ and $E(12, 12)$. The minimum value of $z$ is:

The constraints $-x_{1} + x_{2} \leq 1$,$-x_{1} + 3x_{2} \leq 9$,and $x_{1}, x_{2} \geq 0$ define: