Chapter 2.6, Problem 7CFU

To determine

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

Answer to Problem 7CFU

  $\begin{array}{l}f\left(3,5\right)=-11\text{ Minimum}\\ f\left(5,3\right)=3\text{ Maximum}\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{c}f\left(x,y\right)=3x-4y\\ x-2y\ge -7\\ x+y\ge 8\\ 2x-y\le 7\end{array}$

Calculation:

Graph for the given inequalities is shown below.

  

Solve the function to get maximum and minimum as follows:

  $\begin{array}{l}f\left(3,5\right)=-11\text{ Minimum}\\ f\left(5,3\right)=3\text{ Maximum}\\ f\left(7,7\right)=-7\end{array}$

The coordinate name for the vertices of the polygon set is as follows:

Function	$x-2y\ge -7$	$2x-y\le 7$	$x+y\ge 8$
Coordinate Name	Blue line	Orange line	Green line

Thus, the maximum is $-11$ and 7 is the minimum.