The corner points of the bounded feasible region of a linear programming problem are \((0, 0)\), \((7, 0)\), \((4, 5)\) and \((0, 6)\). For which of the following objective functions does the maximum occur at the point \((4, 5)\) only?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(Z = x + y\)
The maximum of Z on a bounded region is the largest of its corner values. For \(Z = x + y\):
\(Z(0, 0) = 0\)
\(Z(7, 0) = 7\)
\(Z(4, 5) = 9\)
\(Z(0, 6) = 6\)
The largest value 9 occurs only at (4, 5).
Why the other options are incorrect
Option (1) \(Z = 2x + y\) — Incorrect. \(Z(7, 0) = 14\) is larger than \(Z(4, 5) = 13\).
Option (3) \(Z = x + 4y\) — Incorrect. \(Z(4, 5) = 24\) and \(Z(0, 6) = 24\), so the maximum occurs at two corners (and on the edge joining them).
Option (4) \(Z = y\) — Incorrect. The largest y is 6, at (0, 6).