Chapter 3.3, Problem 2AGP

To determine

To find: the coordinate of vertices of feasible regions for the given system of inequalities, $y\le 8,y\ge -x+4\text{and }y\le -x+10$ and find maximum and minimum values of the function $f\left(x,y\right)=-6x+8y$ for this feasible regions.

Answer to Problem 2AGP

Maximum value is $88$ and there is no minimum value.

Explanation of Solution

Given

Given inequalities and function are.

  $\begin{array}{l}y\le 8\\ y\ge -x+4\\ y\le -x+10\\ f\left(x,y\right)=-6x+8y\end{array}$ .

Calculation:

Plotting the graph for given inequalities.

First plotting $y\le 8$ .

Draw vertical lines parallel to x-axis at $y=8$ and shade region below the line because $y\le 8$ as shown in graph below in red colour.

Plotting $y\ge -x+4$ .

Draw graph of the line $y=-x+4$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& 4\\ 4& 0\end{array}$

Shade the region above the line as $y\ge -x+4$ as shown in blue colour.

Plotting $y\le -x+10$ .

Draw graph of the line $y=-x+10$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& 10\\ 10& 0\end{array}$

Shade the region below the line as $y\le -x+10$ as shown in green colour.

  

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

  $\begin{array}{ccc}\left(x,y\right)& -6x+8y& =f\left(x,y\right)\\ \left(2,8\right)& -6\left(2\right)+8\left(8\right)& 52\\ \left(-4,8\right)& -6\left(-4\right)+8\left(8\right)& 88\end{array}$

Therefore, Maximum value is $88$ at $\left(-4,8\right)$ and there is no minimum value.