Which of the following statements is true? A. If the feasible region…

The correct answer is Option (2) → A and C only

In an LPP the optimal value, when it exists, occurs at a corner point of the feasible region. On a bounded region both the maximum and the minimum exist. On an unbounded region Z can grow without limit in some direction, so an optimum may fail to exist.

If the same optimal value occurs at two adjacent corners, every point of the edge joining them is also optimal.

With the usual non-negativity constraints \(x \ge 0, y \ge 0\) and \(a, b \gt 0\):

    \(Z = ax + by \ge 0 \text{ at every feasible point}\)
    \(Z(0, 0) = 0\)

So an unbounded region may have no maximum or minimum, a maximum can occur at two corners, and when the origin is a corner the least value of Z is 0.

Checking the statements

(A) If the feasible region for a LPP is unbounded, maximum or minimum of the objective function Z = ax + by may or may not exist. — Correct. On an unbounded region Z may increase or decrease without limit.

(B) Maximum value of the objective function Z = ax + by in a LPP always occurs at only one corner point of the feasible region. — Incorrect. It can occur at two adjacent corners at once (alternate optimal solutions).

(C) In a LPP, the minimum value of the objective function Z = ax + by \((a, b \gt 0)\) is always 0 if origin is one of the corner points of feasible region. — Correct. Z ≥ 0 on the region and Z(0, 0) = 0.

(D) In a LPP the max value of the objective function Z = ax + by is always finite. — Incorrect. On an unbounded region the maximum may not exist.