State True or False:
If the corner points of the feasible region are (0, 7/3),(2,1),(3, 0) & (0,0) then the maximum value of Z = 4x + 5y is 12 .

Answer

To determine whether the statement is true or false, we need to evaluate the value of \( Z = 4x + 5y \) at the given corner points of the feasible region. The corner points provided are \( (0, \frac{7}{3}) \), \( (2, 1) \), \( (3, 0) \), and \( (0, 0) \).

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

1. **At point \( (0, \frac{7}{3}) \)**:
\[ Z = 4(0) + 5\left(\frac{7}{3}\right) = 0 + \frac{35}{3} = \frac{35}{3} \approx 11.67 \]

2. **At point \( (2, 1) \)**:
\[ Z = 4(2) + 5(1) = 8 + 5 = 13 \]

3. **At point \( (3, 0) \)**:
\[ Z = 4(3) + 5(0) = 12 + 0 = 12 \]

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

### Step 2: Determine the maximum value of \( Z \)

Now, we compare the values of \( Z \) calculated at each corner point:
- At \( (0, \frac{7}{3}) \), \( Z \approx 11.67 \)
- At \( (2, 1) \), \( Z = 13 \)
- At \( (3, 0) \), \( Z = 12 \)
- At \( (0, 0) \), \( Z = 0 \)

The maximum value of \( Z \) among these points is \( 13 \) at the point \( (2, 1) \).

### Conclusion

The statement claims that the maximum value of \( Z \) is \( 12 \). However, we found that the maximum value is actually \( 13 \). Therefore, the statement is **False**.