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

Answer

To find the minimum value of the objective function \( Z = 4x + 6y \) at the given corner points of the feasible region, we will evaluate the function at each corner point. The corner points provided are:

1. \( (0, 2) \)
2. \( (3, 0) \)
3. \( (6, 0) \)
4. \( (6, 8) \)
5. \( (0, 5) \)

### Step 1: Evaluate \( Z \) at each corner point

1. **At \( (0, 2) \)**:
\[ Z = 4(0) + 6(2) = 0 + 12 = 12 \]

2. **At \( (3, 0) \)**:
\[ Z = 4(3) + 6(0) = 12 + 0 = 12 \]

3. **At \( (6, 0) \)**:
\[ Z = 4(6) + 6(0) = 24 + 0 = 24 \]

4. **At \( (6, 8) \)**:
\[ Z = 4(6) + 6(8) = 24 + 48 = 72 \]

5. **At \( (0, 5) \)**:
\[ Z = 4(0) + 6(5) = 0 + 30 = 30 \]

### Step 2: Compare the values of \( Z \)

Now, we will list the values of \( Z \) calculated at each corner point:

- At \( (0, 2) \), \( Z = 12 \)
- At \( (3, 0) \), \( Z = 12 \)
- At \( (6, 0) \), \( Z = 24 \)
- At \( (6, 8) \), \( Z = 72 \)
- At \( (0, 5) \), \( Z = 30 \)

### Step 3: Identify the minimum value

From the calculated values, the minimum value of \( Z \) occurs at the corner points \( (0, 2) \) and \( (3, 0) \), both yielding \( Z = 12 \).

### Conclusion

Thus, the minimum value of \( Z \) occurs at the corner points \( (0, 2) \) and \( (3, 0) \) with a minimum value of \( 12 \).

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