Corner points of the feasible region for an $\operatorname{LPP}$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Let $F = 4x + 6y$ be the objective function. The minimum value of $F$ occurs at $....$

The maximum value of $z=6x+8y$ subject to the constraints $x-y \geq 0$,$x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is:

The minimum value of the objective function $z = 4x + 6y$ subject to the constraints $x + 2y \geq 80$,$3x + y \geq 75$,and $x, y \geq 0$ is:

For the $LPP$,minimize $z = x_{1} + x_{2}$ subject to the constraints $5x_{1} + 10x_{2} \geq 0$,$x_{1} + x_{2} \leq 1$,$x_{2} \leq 4$ and $x_{1}, x_{2} \geq 0$.

$A$ farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$,costing $Rs. 250$ per bag,contains $3$ units of nutritional element $A$,$2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing $Rs. 200$ per bag contains $1.5$ units of nutritional element $A$,$11.25$ units of element $B$,and $3$ units of element $C$. The minimum requirements of nutrients $A$,$B$ and $C$ are $18$ units,$45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

The feasible region represented by the given constraints $2x + 3y \geqslant 12$,$-x + y \leqslant 3$,$x \leqslant 4$,$y \geqslant 3$ is denoted by