The value of an objective function is maximum under linear constraints:

Options

A. at (0,0)
B. at any vertex of feasible region
C. at the centre of feasible region
D. at the vertex which is at maximum distance from (0,0)

Answer

To solve the problem of maximizing an objective function under linear constraints, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Objective Function**:
Let the objective function be defined as \( Z = AX + BY \), where \( A \) and \( B \) are coefficients, and \( X \) and \( Y \) are the variables we want to optimize.

**Hint**: Identify the coefficients of your objective function clearly.

2. **Identify the Constraints**:
Write down the linear constraints that apply to the variables \( X \) and \( Y \). These constraints will typically be in the form of inequalities (e.g., \( a_1X + b_1Y \leq c_1 \)).

**Hint**: Make sure to express all constraints clearly and in the correct format.

3. **Graph the Constraints**:
Plot the constraints on a graph with \( X \) on the horizontal axis and \( Y \) on the vertical axis. Each constraint will form a line, and the area that satisfies all constraints is called the feasible region.

**Hint**: Use a ruler and graph paper for accuracy when plotting the lines.

4. **Determine the Feasible Region**:
Identify the feasible region on the graph, which is the area where all constraints overlap. This region represents all possible solutions that satisfy the constraints.

**Hint**: Shade the feasible region to visualize it better.

5. **Find the Vertices of the Feasible Region**:
Determine the coordinates of the vertices (corners) of the feasible region. These points are where the maximum or minimum values of the objective function will occur.

**Hint**: Use the intersection points of the lines to find the vertices.

6. **Evaluate the Objective Function at Each Vertex**:
Substitute the coordinates of each vertex into the objective function \( Z = AX + BY \) to calculate the value of \( Z \) at each vertex.

**Hint**: Keep track of your calculations to avoid errors.

7. **Identify the Maximum Value**:
Compare the values of \( Z \) obtained from each vertex. The maximum value of \( Z \) will be the highest value calculated among the vertices.

**Hint**: Clearly mark which vertex gives the maximum value for easy reference.

8. **Conclusion**:
State that the maximum value of the objective function occurs at one of the vertices of the feasible region.

**Hint**: Summarize your findings clearly to convey the result effectively.

### Final Answer:
The value of the objective function \( Z \) is maximum at any vertex of the feasible region defined by the linear constraints.