If the corner points of the feasible region for an LPP are (6, 0), (5, 5), (3, 6) and (0, 4) then minimum value of the objective function \(z=2x+3y\) occurs at :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → every point of the line segment joining the points (6, 0) and (4, 0)
By the corner point method, evaluate z = 2x + 3y at each corner:
\(z(6, 0) = 12\)
\(z(5, 5) = 10 + 15 = 25\)
\(z(3, 6) = 6 + 18 = 24\)
\(z(0, 4) = 12\)
The minimum value 12 is reached at two adjacent corners, (6, 0) and (0, 4). Both lie on the line 2x + 3y = 12, which is an edge of the region, so every point of the segment joining (6, 0) and (0, 4) gives z = 12 and is a point of minimum.
Why the other options are incorrect
Option (1) (6, 0) Only — Incorrect. The corner (0, 4) also gives the least value 12, so the minimum is not at (6, 0) only.
Option (2) (0, 4) Only — Incorrect. The corner (6, 0) also gives the least value 12.
Option (3) the mid point of the line segment joining the points (6, 0) and (4, 0) only — Incorrect. The mid point does give 12, but so does every other point of the edge, so "only" is wrong.
Note: It seems that there is a mistake in the original NTA paper, which prints the point (4, 0) in Options (3) and (4) when it should read (0, 4). The point (4, 0) is not a corner point and lies outside the feasible region (there 2x + 3y = 8, less than 12). The question can still be solved, because Option (4) is the only option that describes the whole edge on which the minimum is reached.