Question

In an LPP, if the objective function \[z = ax + by\] has the same maximum value on the two corner points of the feasible region, then the number of points at which \[{z_{\max }}\] occurs is
A.0
B.2
C.Finite
D.Infinite

Answer 1

Hint: In the given question, we have a function which is defined as \[z = ax + by\] . It has been given that the function has the same maximum value on the two corner points of the feasible region. Clearly, the function plots out a geometric figure of a line. So, we have to find the answer pertaining to the line represented by this particular function.

Complete step-by-step answer:
This question is a question of Linear Programming. Linear Programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
In this given question, the given objective function is \[z = ax + by\] .
Here, we have been given that in the feasible region of the geometric figure represented by the given objective function, the corner (or we can say the end) points have the same maximum value.
Now, clearly the geometric figure represented by this objective function is a line.
So, according to the question, since the corner points have the same maximum value, all the points on the line also have the exact maximum value (which here is given to be \[{z_{\max }}\] ).
Thus, the number of points at which \[{z_{\max }}\] occurs is infinite.
So, the correct answer is “Option D”.

Note: So, we saw that in solving questions like this, it is best to first identify the geometric figure or we can say the geometrical shape which the objective function is representing. This way we exactly know what thing we are dealing with and it makes the work quite easier. The corner points mean the extreme points and in this particular question with the specific objective function, each point is the same as the geometric shape which this objective function is representing is a line and that is the reason that this function has the same max value at an infinite number of points.