Chapter 3.7, Problem 50E

To determine

To find: the maximum and minimum of the function using given vertices.

Answer to Problem 50E

The maximum value of the given function is $5$ and the minimum value is $-4$ .

Explanation of Solution

Given:

  $f\left(x,y\right)=y-x$ .

The vertices of the polygon:

  $\left(0,0\right),\left(4,0\right),\left(3,5\right)\text{ and }\left(0,5\right)$ .

Concept used:

By putting the coordinates of each polygon in the function.

The highest value is the maximum and the lowest value is minimum.

Calculation:

The function $f\left(x,y\right)=y-x$ is defined for the polygonal convex set having vertices at:

  $\left(0,0\right),\left(4,0\right),\left(3,5\right)\text{ and }\left(0,5\right)$ are actually the corners points of the polygon, therefore the maximum or minimum value of the given function lies at the Corners.

Now

At $\left(0,0\right)$ the function becomes:

  $\begin{array}{l}f\left(0,0\right)=0-0\\ \Rightarrow 0.\end{array}$

At $\left(4,0\right)$ , the function becomes:

  $\begin{array}{l}f\left(4,0\right)=0-4\\ \Rightarrow -4.\end{array}$

At $\left(3,5\right)$ , the function becomes:

  $\begin{array}{l}f\left(3,5\right)=5-3\\ \Rightarrow 2.\end{array}$

At $\left(0,5\right)$ , the function becomes:

  $\begin{array}{l}f\left(0,5\right)=5-0\\ \Rightarrow 5.\end{array}$

Hence, the maximum value of the given function is $5$ and the minimum value is $-4$ .