Chapter 3.4, Problem 22PPS

To determine

To calculate: The coordinates of the feasible region when the provided system of inequalities are graphed. Also evaluate the minimum and maximum value of the function $f\left(x,y\right)=5x+4y$.

  $\begin{array}{c}y\ge \left|x+1\right|-2\\ 0\le y\le 6\\ -6\le x\le 2\\ x+3y\le 14\end{array}$

Answer to Problem 22PPS

The coordinates of the feasible region are $\left(-3,0\right)\text{,}\left(1,0\right)\text{,}\left(2,1\right)\text{,}\left(2,4\right)\text{,}\left(-4,6\right)\text{,}\left(-6,6\right)\text{ and }\left(-6,3\right)$ . The maximum value of the function is $26$ which occurs at $\left(2,4\right)$ . The minimum value of the function is $-18$ which occurs at $\left(-6,3\right)$.

  

Explanation of Solution

Given information:

The system of inequalities,

  $\begin{array}{c}y\ge \left|x+1\right|-2\\ 0\le y\le 6\\ -6\le x\le 2\\ x+3y\le 14\end{array}$

Calculation:

Consider the system of inequalities,

  $\begin{array}{c}y\ge \left|x+1\right|-2\\ 0\le y\le 6\\ -6\le x\le 2\\ x+3y\le 14\end{array}$

Plot the inequalities $y\ge \left|x+1\right|-2,0\le y\le 6,-6\le x\le 2\text{ and }x+3y\le 14$ on the coordinate plane.

  

The feasible region is the region that is common to all the three inequalities.

Therefore, the region bounded by the vertices $\left(-3,0\right)\text{,}\left(1,0\right)\text{,}\left(2,1\right)\text{,}\left(2,4\right)\text{,}\left(-4,6\right)\text{,}\left(-6,6\right)\text{ and }\left(-6,3\right)$ is the feasible region.

Now, substitute each of the vertices in the function $f\left(x,y\right)=5x+4y$ to compute the maximum and minimum value of the function.

Value of the function at the vertex $\left(-3,0\right)$ ,

  $f\left(-3,0\right)=-15$

Value of the function at the vertex $\left(1,0\right)$ ,

  $f\left(1,0\right)=5$

Value of the function at the vertex $\left(2,1\right)$ ,

  $f\left(2,1\right)=14$

Value of the function at the vertex $\left(2,4\right)$ ,

  $f\left(2,4\right)=26$

Value of the function at the vertex $\left(-4,6\right)$ ,

  $f\left(-4,6\right)=4$

Value of the function at the vertex $\left(-6,6\right)$ ,

  $f\left(-6,6\right)=-6$

Value of the function at the vertex $\left(-6,3\right)$ ,

  $f\left(-6,3\right)=-18$

Therefore, the maximum value of the function is $26$ which occurs at $\left(2,4\right)$ . The minimum value of the function is $-18$ which occurs at $\left(-6,3\right)$.