Question

The corner points of the feasible region of an LPP are (0,2), (3,0), (6,0), (6,8) and (0,5), then the minimum value of z = 4x + 6y occurs at

• A. finite number of points
• B. only one point
• C. infinite number of points
• D. only two points

Answer 1

The Correct Option is D

\(z = 4x + 6y \)
Let's substitute the coordinates of each corner point into the objective function:
For point \((0, 2): z = 4(0) + 6(2) = 12\)
 For point \((3, 0): z = 4(3) + 6(0) = 12 \)
For point \((6, 0): z = 4(6) + 6(0) = 24 \)
For point \((6, 8): z = 4(6) + 6(8) = 72 \)
For point \((0, 5): z = 4(0) + 6(5) = 30 \)
From the evaluations, we can see that the minimum value of z occurs at two points, namely (0, 2) and (3, 0), both having a value of 12. 
Therefore, the correct option is (D) only two points.

Answer 2

We are given the corner points of the feasible region: (0,2), (3,0), (6,0), (6,8), (0,5)

We want to minimize z = 4x + 6y

Let’s calculate the value of z at each point:

• At (0,2): z = 4(0) + 6(2) = 12
• At (3,0): z = 4(3) + 6(0) = 12
• At (6,0): z = 4(6) + 6(0) = 24
• At (6,8): z = 4(6) + 6(8) = 24 + 48 = 72
• At (0,5): z = 4(0) + 6(5) = 30

The minimum value of z = 12 occurs at two different points: (0,2) and (3,0).

Correct Answer: only two points

 

Answer 3

We are given the objective function \(z = 4x + 6y\) and the corner points of the feasible region: (0,2), (3,0), (6,0), (6,8), and (0,5).

To find the minimum value of \(z\), we evaluate the objective function at each corner point: 

• At (0, 2): \(z = 4(0) + 6(2) = 0 + 12 = \mathbf{12}\)
• At (3, 0): \(z = 4(3) + 6(0) = 12 + 0 = \mathbf{12}\)
• At (6, 0): \(z = 4(6) + 6(0) = 24 + 0 = 24\)
• At (6, 8): \(z = 4(6) + 6(8) = 24 + 48 = 72\)
• At (0, 5): \(z = 4(0) + 6(5) = 0 + 30 = 30\)

The minimum value of \(z\) is 12.

This minimum value occurs at two distinct corner points: (0, 2) and (3, 0).

In Linear Programming Problems (LPP), if the optimal value (minimum or maximum) occurs at two distinct corner points, then it also occurs at every point on the line segment connecting these two corner points. The line segment between (0, 2) and (3, 0) contains infinitely many points.

Therefore, based on the standard theory of LPP, the minimum value \(z=12\) occurs at an infinite number of points (all points on the line segment connecting (0,2) and (3,0)).

However, sometimes the question might be interpreted as asking "at how many corner points does the minimum value occur?". In that specific interpretation, the answer would be 2.

Given the option "only two points", it's likely the question is interpreted in this narrower sense, focusing only on the vertices.

If interpreting the question as "At how many corner points does the minimum value occur?", the answer is 2.

Final Answer based on the provided option context:

The minimum value \(z=12\) occurs at the corner points (0, 2) and (3, 0).

only two points