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. The minimum value of F occurs at:

Options

A. (0,2) only
B. (3,0) only
C. The mid-point of the line segment joining the points (0,2) and (3,0) only
D. Any point on the line segment joining the points (0,2) and (3,0)

Answer

To find the minimum value of the objective function \( F = 4x + 6y \) at the given corner points of the feasible region, we will evaluate the function at each of the 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 \( (0, 5) \)**:
\[ F(0, 5) = 4(0) + 6(5) = 0 + 30 = 30 \]

6. **Compare the values**:
- \( F(0, 2) = 12 \)
- \( F(3, 0) = 12 \)
- \( F(6, 0) = 24 \)
- \( F(6, 8) = 72 \)
- \( F(0, 5) = 30 \)

The minimum value of \( F \) occurs at the points \( (0, 2) \) and \( (3, 0) \), both yielding a value of \( 12 \).

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

---