If $z = ax + by$ where $a, b > 0$ subject to the constraints $x \leq 2, y \leq 2, x + y \geq 3, x \geq 0, y \geq 0$ has a minimum value at $(2, 1)$ only,then...

$A$ manufacturer makes two types of toys $A$ and $B$. Three machines are needed for this purpose and the time (in $minutes$) required for each toy on the machines is given below:

Types of Toys	Machine-$I$	Machine-$II$	Machine-$III$
$A$	$12$	$18$	$6$
$B$	$6$	$0$	$9$

Each machine is available for a maximum of $6 \, hours$ $(360 \, minutes)$ per day. If the profit on each toy of type $A$ is $Rs. \, 7.50$ and that on each toy of type $B$ is $Rs. \, 5$,find the number of toys of each type that should be manufactured in a day to get maximum profit.

The shaded region in the following figure is the solution set of the inequations:

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is

The objective function $z=x_1+x_2$,subject to $x_1+x_2 \leq 10, -2x_1+3x_2 \leq 15, x_1 \leq 6, x_1, x_2 \geq 0$,has a maximum value at:

The maximum value of $z=x+y$,subjected to $x+y \leq 10, 5x+3y \geq 15, x \leq 6, x, y \geq 0$,