Linear Programming

Step 1: Understanding the Objective Function.
The objective function in a linear programming problem is always a linear function. It is a mathematical function that needs to be maximized or minimized, subject to a set of linear constraints. The form of the objective function is always linear, hence option (B) is correct. Step 2: Conclusion.
Thus, the objective function in linear programming is always linear, corresponding to option (B).

Step 1: Understanding the objective function in Linear Programming Problems (L.P.P).
In Linear Programming Problems (L.P.P), the objective function is a key component. It represents the function that needs to be either maximized or minimized depending on the problem's requirements. The objective function is expressed as a linear combination of decision variables, and the goal is to find the values of these variables that optimize the objective function while satisfying the constraints. Step 2: Why the statement is true.
The statement is correct because the purpose of an L.P.P is to optimize the objective function subject to constraints. Whether the goal is to maximize profit, minimize cost, or achieve some other optimal condition, the objective function is the function that defines what is being optimized. Conclusion:
Thus, the statement is True. The objective function is indeed the function to be optimized in a Linear Programming Problem.

Step 1: Graphing the constraints.
The constraints are: These inequalities describe a region in the first quadrant of the -plane. The line intersects the -axis at and the -axis at .
Step 2: Maximizing .
To maximize , we evaluate at the vertices of the feasible region. The vertices are , , and .
- At ,
- At ,
- At ,
Step 3: Conclusion.
Thus, the maximum value of is , which occurs at .