If the objective function for an L.P.P. is `Z=3x-4y` and the corner points for the bounded feasible region are (0, 0), (5, 0), (6, 5), (6, 8), (4, 10) and (0, 8), then the maximum value of Z occurs at

Options

A. (0, 0)
B. (0, 8)
C. (5, 0)
D. (4, 10)

Answer

To solve the linear programming problem (L.P.P.) and find the maximum value of the objective function \( Z = 3x - 4y \) at the given corner points, we will evaluate the function at each corner point.

### Step-by-Step Solution:

1. **Identify the Objective Function**:
The objective function is given as:
\[ Z = 3x - 4y \]

2. **List the Corner Points**:
The corner points provided are:
\[ (0, 0), (5, 0), (6, 5), (6, 8), (4, 10), (0, 8) \]

3. **Evaluate the Objective Function at Each Corner Point**:

- For the point \( (0, 0) \):
\[ Z = 3(0) - 4(0) = 0 \]

- For the point \( (5, 0) \):
\[ Z = 3(5) - 4(0) = 15 \]

- For the point \( (6, 5) \):
\[ Z = 3(6) - 4(5) = 18 - 20 = -2 \]

- For the point \( (6, 8) \):
\[ Z = 3(6) - 4(8) = 18 - 32 = -14 \]

- For the point \( (4, 10) \):
\[ Z = 3(4) - 4(10) = 12 - 40 = -28 \]

- For the point \( (0, 8) \):
\[ Z = 3(0) - 4(8) = 0 - 32 = -32 \]

4. **Compare the Values of Z**:
Now we will compare the values of \( Z \) obtained from each corner point:
- \( Z(0, 0) = 0 \)
- \( Z(5, 0) = 15 \)
- \( Z(6, 5) = -2 \)
- \( Z(6, 8) = -14 \)
- \( Z(4, 10) = -28 \)
- \( Z(0, 8) = -32 \)

5. **Determine the Maximum Value**:
The maximum value of \( Z \) is:
\[ \text{Maximum } Z = 15 \text{ at the point } (5, 0) \]

### Conclusion:
The maximum value of \( Z \) occurs at the corner point \( (5, 0) \).