Chapter 2.7, Problem 11E

To determine

To graph: the system of inequalities, find the maximum values of $f\left(x,y\right)$ , and state whether this situation is infeasible, has alternate optimal solutions, or is unbounded.

Answer to Problem 11E

The situation is alternate optimal solutions.

Explanation of Solution

Given information:

Assume that $x\ge 0$ and $y\ge 0$ .

  $\begin{array}{c}4x+3y\ge 12\\ y\le 3\\ x\le 4\\ f\left(x,y\right)=3+3y\end{array}$

Calculation:

The graph for the given inequalities is as follows:

  

The vertices of the boundary region are $\left(3,0\right),\left(4,0\right),\left(\frac{3}{4},3\right)$ and $\left(4,3\right)$ . Put the values in $f\left(x,y\right)$ and solve to find the maximum value as follows:

  $\begin{array}{c}f\left(3,0\right)=3+3\left(0\right)\\ =3\\ f\left(4,0\right)=3+3\left(0\right)\\ =3\\ f\left(\frac{3}{4},3\right)=3+3\left(3\right)\\ =12\\ f\left(4,3\right)=3+3\left(3\right)\\ =12\end{array}$

There is more than one point with a maximum value. So, this problem has alternate optimal solutions.