Question

The vertices of a feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$write a function that satisfy each condition.

a. A is the maximum and B is minimum

Short answer

a.The function $f=y-x$satisfy the condition

b. The function$f=y-x$satisfy the condition

c. The function$g=x+y$satisfy the condition

d. The function$h=x-y$satisfy the condition

e.The function$t=x+2y$satisfy the condition

Step-by-step solution

Part a. Step 1. Find the feasible solution which satisfies that A is maximum 

The vertices of the feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$

Step 2. Determination of maximum and minimum point

Let the function be$f(x,y)=x+y$

Then, at point $A\left(1,2\right)$

$f(x,y)=-1+2=1$

At point $B\left(5,2\right)$

$\begin{array}{c}f(x,y)=-5+2\\ =-3\end{array}$

b. c is the maximum and B is the minimum

Part b. Step 1. Find the feasible solution which satisfies that C is maximum and B is minimum

Find the feasible solution which satisfies that A is maximum and B is minimum.

The vertices of the feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$

Step 2. Determination of maximum and minimum point

Let the function be $f(x,y)=x+y$

At point $C\left(1,4\right)$

$f(x,y)=4-1=3$

At point $B\left(5,2\right)$

$f(x,y)=2-5=-3$

c. B is the maximum and A is minimum

Part c. Step 1. Find the feasible solution which satisfies B is maximum and A is minimum

Find the feasible solutions which satisfies that A is maximum and B is minimum.

The vertices of the feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$

Step 2. Determination of maximum and minimum point

Let the function function be$g(x,y)=x+y$

Then, at point $B\left(5,2\right)$

$g(x,y)=5+2=7$

At point $A\left(1,2\right)$

$g(x,y)=1+2=3$

d. A is the maximum and C is minimum

Part d. Step 1. Find the feasible solution which satisfies A is maximum and C is minimum

Find the feasible solution which satisfies that A is maximum and B is minimum.

The vertices of the feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$

Step 2. Determination of maximum and minimum point

Let the function be$h(x,y)=x-y$

Then, at point $A\left(1,2\right)$

$h(x,y)=1-2=-1$

At point $C\left(1,4\right)$

$h(x,y)=1-4=-3$

e. B and C are both maxima and A is minimum

Part e. Step 1. Find the feasible solution which satisfies B and C both are maxima and A is the minimum

Find the feasible solution which satisfies that A is maximum and B is minimum.

The vertices of the feasible region are $A\left(1,2\right),B\left(5,2\right)andC\left(1,4\right)$

Step 2. Determination of maximum and minimum point

Let the function function be$t(x,y)=x+2y$

Then, at point $A\left(1,2\right)$

$\begin{array}{c}h(x,y)=1+2\left(2\right)\\ =1+4\\ =5\end{array}$

At point $B\left(5,2\right)$

$\begin{array}{c}t(x,y)=5+2\left(2\right)\\ =5+4\\ =9\end{array}$

At point $C\left(1,4\right)$

$\begin{array}{c}t(x,y)=1+2\left(4\right)\\ =1+8\\ =9\end{array}$