Chapter 2.6, Problem 22E

To determine

To find: the maximum and minimum values for $f\left(x,y\right)$ .

Answer to Problem 22E

Maximum value is 11.

Minimum value is $-5$ .

Explanation of Solution

Given information:

  $\begin{array}{c}f\left(x,y\right)=-2x+y+5\\ 2\le y\le 8\\ x\ge 1\\ 2x+y+2\le 16\\ y\ge 5-x\end{array}$

Calculation:

Graph for the given inequalities is shown below.

  

The vertices of the polygon boundary are $\left(1,8\right),\left(1,4\right),\left(3,8\right),\left(3,2\right)$ and $\left(6,2\right)$ . The maximum and minimum value of $f\left(x,y\right)$ will occur at these points.

Solve the function to get maximum and minimum as follows:

  $\begin{array}{c}f\left(1,8\right)=-2\left(1\right)+8+5\\ =-2+8+5\\ =11\\ \\ f\left(1,4\right)=-2\left(1\right)+4+5\\ =-2x+4+5\\ =7\\ \\ f\left(3,8\right)=-2\left(3\right)+8+5\\ =-6+8+5\\ =7\\ \\ f\left(3,2\right)=-2\left(3\right)+2+5\\ =-6+2+5\\ =1\\ \\ f\left(6,2\right)=-2\left(6\right)+2+5\\ =-12+2+5\\ =-5\end{array}$

Thus, the maximum and minimum value of $f\left(x,y\right)$ is 11 and $-5$ respectively.