Chapter 7.6, Problem 33E

To determine

To find: the minimum and maximum values of the given objective function and the points where the values occur.

Answer to Problem 33E

The maximum value of $z=4$ and this occurs on the line $x=0$ and $y=1$ .

The minimum value of $z=0$ and this occurs at $x=0$ and $y=0$ .

Explanation of Solution

Given:

The given objective function

  $z=3x+4y$

Constraints:

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 1\\ 2x+y\ge 4\end{array}$

Calculation:

Objective function

  $z=3x+4y$

Subjected to the following constraints:

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 1\\ 2x+y\ge 4\end{array}$

The area bounded by the constraints are as shown below:

  

At the three vertices of the region formed by the constraints the objective function has the following values:

At $\left(0,0\right):z=3\left(0\right)+4\left(0\right)=0$ Minimum value of $z$

At $\left(0,1\right):z=3\left(0\right)+4\left(1\right)=4$ Maximum value of $z$

At $\left(1,0\right):z=3\left(1\right)+4\left(0\right)=3$

So, the maximum value of $z=4$ and this occurs on the line $x=0$ and $y=1$ .

The minimum value of $z=0$ and this occurs at $x=0$ and $y=0$ .

Here is constrain $2x+y\le 4$ is insignificant.