Chapter 3.3, Problem 2CYU

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 2CYU

The vertices of the feasible region are,

  $\left(-3,3\right),\left(3,-3\right)and\left(1,3\right)$

The maximum value of the given function is 20 at point $\left(1,3\right)$ and minimum value is $-12$ at point $\left(-3,3\right)$ .

Explanation of Solution

Given information:

Given system of inequalities,

  $\begin{array}{l}y\le -3x+6\\ -y\le x\\ y\le 3\end{array}$

Given function

  $f\left(x,y\right)=8x+4y$

Calculation:

Consider the given function

  $f\left(x,y\right)=8x+4y$

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,

  $\left(-3,3\right),\left(3,-3\right)and\left(1,3\right)$

Compute the value of the given function $f\left(x,y\right)=8x+4y$ at given vertex,

At point $\left(-3,3\right)$ .

  $\begin{array}{c}f\left(-3,3\right)=8\times \left(-3\right)+4\times 3\\ =-24+12\\ =-12\end{array}$

At point $\left(3,-3\right)$ ,

  $\begin{array}{c}f\left(3,-3\right)=8\times \left(3\right)+4\times \left(-3\right)\\ =24-12\\ =12\end{array}$

At point $\left(1,3\right)$ ,

  $\begin{array}{c}f\left(1,3\right)=8\times \left(1\right)+4\times \left(3\right)\\ =8+12\\ =20\end{array}$

Therefore, the maximum value of the given function is 20 at point $\left(1,3\right)$ and minimum value is $-12$ at point $\left(-3,3\right)$ .