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 objective function of a Linear Programming Problem $(L.P.P.)$ defined over a convex set attains its optimum value at

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:

Value of the objective function $Z = -50x + 20y$ subject to the constraints $2x - y \geq -5$,$3x + y \geq 3$,$2x - 3y \leq 12$,$x \geq 0$,$y \geq 0$. The corner points of the feasible region are $(0, 5)$,$(0, 3)$,$(1, 0)$,and $(6, 0)$. At which point is the value of $Z$ minimum?

The corner points of the feasible region are $A(0,0)$,$B(16,0)$,$C(8,16)$,and $D(0,24)$. The minimum value of the objective function $z = 300x + 190y$ is . . . . . . .

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