If LPP has optimal solution at two point then,

Options

A. LPP will give unique solution
B. LPP will give two solutions.
C. LPP will give infinite solutions.
D. LPP will not give any convex set.

Answer

To solve the question, we need to analyze the implications of having an optimal solution at two points in a Linear Programming Problem (LPP).

### Step-by-Step Solution:

1. **Understanding LPP**: Linear Programming Problems involve maximizing or minimizing a linear objective function subject to linear constraints. The feasible region formed by these constraints is typically a convex polygon (in two dimensions).

**Hint**: Remember that the feasible region is formed by the intersection of constraints, and it is convex.

2. **Optimal Solutions**: In LPP, an optimal solution is a point in the feasible region where the objective function reaches its maximum or minimum value.

**Hint**: An optimal solution can occur at the vertices (corner points) of the feasible region.

3. **Multiple Optimal Solutions**: If there are multiple optimal solutions, it means that the objective function has the same value at more than one point in the feasible region. This often occurs when the objective function is parallel to a constraint line over a segment of the feasible region.

**Hint**: Think about how parallel lines can indicate multiple optimal points.

4. **Infinitely Many Solutions**: If there are two distinct points where the objective function is optimal, it implies that all points along the line segment connecting these two points are also optimal. This leads to the conclusion that there are infinitely many optimal solutions.

**Hint**: Visualize the line segment between two optimal points; every point on this segment is also an optimal solution.

5. **Conclusion**: Therefore, if an LPP has optimal solutions at two points, it indicates that there are infinitely many optimal solutions.

**Final Answer**: The correct option is that if LPP has optimal solutions at two points, then LPP will give infinitely many solutions.

### Summary:
- If LPP has optimal solutions at two points, it leads to infinitely many optimal solutions due to the nature of linear functions and the convexity of the feasible region.