Which of the following is correct?

The correct answer is Option (4) → If a LPP admits optimal solution at two points, then it has optimal solution at infinite number of points.

The feasible region of an LPP is convex, and the objective function Z = ax + by is linear. If Z has the same optimal value at two points P and Q, then every point of the segment PQ lies in the region and gives Z the same value, so there are infinitely many optimal points.

Checking each option:

Option (1) Every LPP admits an optimal solution. — Incorrect. The feasible region may be empty (no point satisfies all the constraints), or unbounded so that Z can be made as large as we like; then there is no optimal solution.

Option (2) Every LPP admits a unique solution. — Incorrect. When the line Z = constant is parallel to an edge of the region, the optimal value is reached at two corners and at every point between them, so the solution is not unique.

Option (3) The optimal value does not occur at a corner point of the feasible region only. — Incorrect. By the corner point theorem, an optimal value, when it exists, is always reached at a corner point; it is reached at other points as well only in the special case of alternate optimal solutions, so the statement does not hold in general.

Option (4) If a LPP admits optimal solution at two points, then it has optimal solution at infinite number of points. — Correct. Every point of the segment joining the two optimal points gives the same optimal value.