Corner points of the feasible region for an LPP are (0,2),(3,0),(6,0),(6,8), and (0,5). Let F-4x+6y be the objective function. Determine the minimum valur of F occurs at

Options

A. only (0,2)
B. only (3,0)
C. the mid point of the line segment joining the points (0,2) and (3,0)
D. any point of the line segemet joining the points (0,2) and (3,0)

Explanation

To determine the minimum value of the objective function \( F = 4x + 6y \) at the corner points of the feasible region, we will evaluate \( F \) at each of the given corner points: \( (0, 2) \), \( (3, 0) \), \( (6, 0) \), \( (6, 8) \), and \( (0, 5) \).

### Step-by-Step Solution:

1. **Evaluate F at (0, 2)**:
\[ F(0, 2) = 4(0) + 6(2) = 0 + 12 = 12 \]

2. **Evaluate F at (3, 0)**:
\[ F(3, 0) = 4(3) + 6(0) = 12 + 0 = 12 \]

3. **Evaluate F at (6, 0)**:
\[ F(6, 0) = 4(6) + 6(0) = 24 + 0 = 24 \]

4. **Evaluate F at (6, 8)**:
\[ F(6, 8) = 4(6) + 6(8) = 24 + 48 = 72 \]

5. **Evaluate F at (0, 5)**:
\[ F(0, 5) = 4(0) + 6(5) = 0 + 30 = 30 \]

### Summary of Values:
- At \( (0, 2) \), \( F = 12 \)
- At \( (3, 0) \), \( F = 12 \)
- At \( (6, 0) \), \( F = 24 \)
- At \( (6, 8) \), \( F = 72 \)
- At \( (0, 5) \), \( F = 30 \)

### Conclusion:
The minimum value of \( F \) occurs at the points \( (0, 2) \) and \( (3, 0) \), both giving \( F = 12 \).