The corner points of the feasible region are `(4, 2), (5,0), (4,1)` and `(6,0)` then the point of minimum `z = 3.5x + 2y= 16` is at

Options

A. (4,2)
B. (5,0)
C. (6,0)
D. (4,1)

Answer

To find the point of minimum \( z = 3.5x + 2y \) given the corner points of the feasible region, we will evaluate \( z \) at each of the corner points: \( (4, 2) \), \( (5, 0) \), \( (4, 1) \), and \( (6, 0) \).

### Step-by-Step Solution:

1. **Evaluate \( z \) at the point \( (4, 2) \)**:
\[ z = 3.5(4) + 2(2) = 14 + 4 = 18 \]

2. **Evaluate \( z \) at the point \( (5, 0) \)**:
\[ z = 3.5(5) + 2(0) = 17.5 + 0 = 17.5 \]

3. **Evaluate \( z \) at the point \( (4, 1) \)**:
\[ z = 3.5(4) + 2(1) = 14 + 2 = 16 \]

4. **Evaluate \( z \) at the point \( (6, 0) \)**:
\[ z = 3.5(6) + 2(0) = 21 + 0 = 21 \]

5. **Compare the values of \( z \)**:
- At \( (4, 2) \), \( z = 18 \)
- At \( (5, 0) \), \( z = 17.5 \)
- At \( (4, 1) \), \( z = 16 \)
- At \( (6, 0) \), \( z = 21 \)

6. **Determine the minimum value**:
The minimum value of \( z \) is \( 16 \) which occurs at the point \( (4, 1) \).

### Conclusion:
The point of minimum \( z \) is at \( (4, 1) \).

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