The minimum value of the objective function Z=x+2y
Subject to the constraints,
`2x+ y ge 3 , x +2y ge 6 , x, y ge 0` occurs

Options

A. at every point on the line x + 2y = 6
B. at every point on the line 2x + y = 3
C. at every point on the line x + 2y = 3 d
D. at every point on the line 2x + y = 6

Explanation

To solve the problem of finding the minimum value of the objective function \( Z = x + 2y \) subject to the constraints \( 2x + y \geq 3 \), \( x + 2y \geq 6 \), and \( x, y \geq 0 \), we will follow these steps:

### Step 1: Write the constraints in standard form
The constraints given are:
1. \( 2x + y \geq 3 \)
2. \( x + 2y \geq 6 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

We can rewrite the inequalities as:
1. \( 2x + y = 3 \)
2. \( x + 2y = 6 \)

### Step 2: Find the intersection points of the lines
To find the feasible region, we first need to determine the points where the lines intersect.

**For the first line \( 2x + y = 3 \):**
- When \( x = 0 \):
\[ 2(0) + y = 3 \implies y = 3 \quad \text{(Point A: (0, 3))} \]
- When \( y = 0 \):
\[ 2x + 0 = 3 \implies x = \frac{3}{2} \quad \text{(Point B: (1.5, 0))} \]

**For the second line \( x + 2y = 6 \):**
- When \( x = 0 \):
\[ 0 + 2y = 6 \implies y = 3 \quad \text{(Point C: (0, 3))} \]
- When \( y = 0 \):
\[ x + 0 = 6 \implies x = 6 \quad \text{(Point D: (6, 0))} \]

### Step 3: Identify the feasible region
Now, we plot the lines on the coordinate system and identify the feasible region that satisfies all constraints. The feasible region is where both inequalities hold true.

### Step 4: Determine the vertices of the feasible region
From the points calculated:
- Point A: (0, 3)
- Point B: (1.5, 0)
- Point D: (6, 0)

The feasible region is bounded by these points and the axes.

### Step 5: Evaluate the objective function at each vertex
Now, we will evaluate the objective function \( Z = x + 2y \) at each vertex of the feasible region:

1. At Point A (0, 3):
\[ Z = 0 + 2(3) = 6 \]

2. At Point B (1.5, 0):
\[ Z = 1.5 + 2(0) = 1.5 \]

3. At Point D (6, 0):
\[ Z = 6 + 2(0) = 6 \]

### Step 6: Find the minimum value
From the evaluations:
- At (0, 3), \( Z = 6 \)
- At (1.5, 0), \( Z = 1.5 \)
- At (6, 0), \( Z = 6 \)

The minimum value of \( Z \) occurs at the point (1.5, 0) where \( Z = 1.5 \).

### Final Answer
The minimum value of the objective function \( Z = x + 2y \) occurs at the point \( (1.5, 0) \) with a minimum value of \( Z = 1.5 \).

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