Chapter 3.3, Problem 6CYU

To determine

To sketch: the graph given system of inequalities and to name the coordinates of the vertices of the feasible region, and to find the maximum and minimum values of the given function for this region.

Answer to Problem 6CYU

The vertices of the feasible region are,

  $\left(-6,-6\right)and\left(-2.5,-13\right)$

There is “No minimum value” of the given function and maximum value is $39.5$ at point $\left(-2.5,-13\right)$

Explanation of Solution

Given information:

Given system of inequalities,

  $\begin{array}{l}y\le 2x+6\\ y\ge 2x-8\\ y\ge -2x-18\end{array}$

Given function

  $f\left(x,y\right)=5x-4y$

Calculation:

Consider the given function

  $f\left(x,y\right)=5x-4y$

Graph given system of inequalities on the same graph paper as follows using MAPLE as given below:

  

From the graph, the coordinates of the vertices of the feasible region are,

  $\left(-6,-6\right)and\left(-2.5,-13\right)$

Compute the value of the given function $f\left(x,y\right)=5x-4y$ at given vertex,

At point $\left(-6,-6\right)$ .

  $\begin{array}{c}f\left(-6,-6\right)=5\times \left(-6\right)-4\times \left(-6\right)\\ =-30+24\\ =-6\end{array}$

At point $\left(-2.5,-13\right)$ ,

  $\begin{array}{c}f\left(-2.5,-13\right)=5\times \left(-2.5\right)-4\times \left(-13\right)\\ =-12.5+52\\ =39.5\end{array}$

As the feasible region is unbounded, take any point from the feasible region and compute the value of function at the point. For example $\left(0,5\right)$ ,

  $\begin{array}{c}f\left(0,5\right)=5\times \left(0\right)-4\times \left(5\right)\\ =0-20\\ =-20\end{array}$

The value of function at point $\left(0,5\right)$ is greater than the maximum value $\left(-6\right)$ of the function at the vertices, there is no minimum value of the given function.

Hence, there is “No minimum value” of the given function and maximum value is $39.5$ at point $\left(-2.5,-13\right)$