Chapter 3.3, Problem 9PPS

To determine

To Find: the coordinate of vertices of feasible regions for the given system of inequalities, $2\ge x\ge -3,y\ge -2x-6\text{ and }4y\le 2x+32$ and find maximum and minimum values of the function $f\left(x,y\right)=-4x-9y$ for this feasible regions.

Answer to Problem 9PPS

  $82$ and $-89$ .

Explanation of Solution

Given

Given inequalities are.

  $\begin{array}{l}2\ge x\ge -3\\ y\ge -2x-6\\ 4y\le 2x+32\\ f\left(x,y\right)=-4x-9y\end{array}$ .

Calculation:

Plotting the graph for given inequalities.

Plotting $2\ge x\ge -3$ .

Draw two vertical lines parallel to y-axis passing through $x=2\text{and}x=-3$ ,shade the region between the lines as shown in red colour.

Plotting $y\ge -2x-6$ .

Draw graph of the line $y=-2x-6$ for the coordinates.

  $\begin{array}{cc}x& y\\ 0& -6\\ -3& 0\end{array}$

Shade the region above the line as $y\ge -2x-6$ .

On simplifying the $4y\le 2x+32$ .

  $\begin{array}{l}4y\le 2x+32\\ y\le \frac{1}{2}x+8\end{array}$

Plotting $y\le \frac{1}{2}x+8$ .

Draw graph of the line $y=\frac{1}{2}x+8$ for the coordinates.

  $\begin{array}{cc}x& y\\ 0& 8\\ -16& 0\end{array}$

Shade the region below the line as $y\le \frac{1}{2}x+8$ .

  

Coordinates of the vertices of feasible region are given in table below.

  $\begin{array}{ccc}\left(x,y\right)& -4x-9y& =f\left(x,y\right)\\ \left(-3,6.5\right)& -4\left(-3\right)-9\left(6.5\right)& -46.5\\ \left(2,9\right)& -4\left(2\right)-9\left(9\right)& -89\\ \left(2,-10\right)& -4\left(2\right)-9\left(-10\right)& 82\\ \left(-3,0\right)& -4\left(-3\right)-9\left(0\right)& 12\end{array}$

Therefore, Maximum value is $82$ at $\left(2,-10\right)$ and minimum value is $-89$ at $\left(2,9\right)$ .