The minimum value of \(Z = 3x + 4y\) subject to the constraints \(x + y \le 5\), \(x \ge 1\), \(y \ge 1\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 7
The region lies to the right of x = 1, above y = 1 and below x + y = 5. It is a bounded triangle.
Its corner points:
x = 1 with y = 1: (1, 1)
y = 1 with x + y = 5: (4, 1)
x = 1 with x + y = 5: (1, 4)
Values of Z = 3x + 4y:
\(Z(1, 1) = 3 + 4 = 7\)
\(Z(4, 1) = 12 + 4 = 16\)
\(Z(1, 4) = 3 + 16 = 19\)
The minimum is 7, at (1, 1).
Why the other options are incorrect
Option (1) 0 — Incorrect. This is Z at (0, 0), which is not feasible: it breaks \(x \ge 1\) and \(y \ge 1\).
Option (2) 16 — Incorrect. This is Z at the corner (4, 1), which is larger than 7.
Option (3) 19 — Incorrect. This is Z at the corner (1, 4); it is the maximum of Z, not the minimum.