Which of the following statements is correct?

The correct answer is Option (3) → If a linear programming problem has two optimal solutions, then it has infinitely many optimal solutions.

If the objective function Z = ax + by takes its optimal value at two points, it takes the same value at every point of the line segment joining them. This is because Z is linear and the feasible region is convex, so the whole segment lies in the region. A segment has infinitely many points.

Option (1) Every linear programming problem has at least one optimal solution. — Incorrect. A problem whose constraints have no common point (empty feasible region), or whose objective grows without limit on an unbounded region, has no optimal solution.

Option (2) Every linear programming problem has a unique optimal solution. — Incorrect. When the objective line is parallel to an edge of the region, every point of that edge is optimal.

Option (3) If a linear programming problem has two optimal solutions, then it has infinitely many optimal solutions. — Correct. Every point of the segment joining the two solutions is also optimal.

Option (4) The feasible region of a linear programming problem is always bounded. — Incorrect. For example, the region \(x+y \ge 2\), \(x, y \ge 0\) extends without limit.