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:

For a linear programming problem,the objective function is $Z = 3x + 2y$. If the corner points of the bounded feasible region are $(12, 0)$,$(4, 2)$,$(1, 5)$,and $(1, 10)$,then the maximum value of $Z$ is . . . . . . .

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 coordinates of the corner points of the bounded feasible region are $(0, 0), (0, 40), (20, 40), (60, 20), (60, 0)$. The maximum of the objective function $z = 40x + 30y$ is . . . . . . .

The corner points of the feasible region determined by the system of linear constraints are $(0,10), (5,5), (15,15), (0,20)$. Let $z = px + qy$,where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points $(15,15)$ and $(0,20)$ is $\ldots \ldots$

The objective function of a Linear Programming Problem $(LPP)$ defined over a convex set attains its optimum value at: