Corner points of the feasible region determined by the system of linear constraints are $(0,3)$,$(1,1)$ and $(3,0)$. Let $Z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the minimum value of $Z$ occurs at $(3,0)$ and $(1,1)$ is . . . . . . .

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:

The coordinates of the corner points of the bounded feasible region are $(0,10), (5,5), (15,15)$,and $(0,20)$. The maximum value of the objective function $Z = 10x + 20y$ is:

The corner points of the feasible region determined by the system of linear inequalities $2x + y \leq 10$,$x + 3y \leq 15$,$x, y \geq 0$ are $(0,0)$,$(5,0)$,$(3,4)$,and $(0,5)$. Let $Z = qx + py$ where $p, q > 0$. The condition on $p$ and $q$ such that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is:

The corner points of the feasible region of an $LPP$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Then the minimum value of $z = 4x + 6y$ occurs at:

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