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

$A$ fruit grower can use two types of fertilizer in his garden,brand $P$ and brand $Q$. The amounts (in $kg$) of nitrogen,phosphoric acid,potash,and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240\,kg$ of phosphoric acid,at least $270\,kg$ of potash and at most $310\,kg$ of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden,how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden (in $,kg$)?

	Brand $P$ ($kg$ per bag)	Brand $Q$ ($kg$ per bag)
Nitrogen	$3$	$3.5$
Phosphoric acid	$1$	$2$
Potash	$3$	$1.5$
Chlorine	$1.5$	$2$

Let $x$ and $y$ be optimal solutions of a Linear Programming $(LP)$ problem. Then,which of the following is true?

In the following figure,the feasible region (shaded) for a $LPP$ is shown. Determine the maximum and minimum value of $Z=x+2y$.

The feasible region for an $LPP$ is shown in the figure. Let $z=3x-4y$ be the objective function. The maximum value of $z$ is $....$

In a Linear Programming Problem ($L$.$P$.$P$.), the corner points of the feasible region determined by the constraints $3x - y \geq 6$, $x \leq 3$, $y \leq 2$, $y \geq 0$, and $x \geq 0$ are: