What is the optimal value of the objective function in the given linear programming scenario?

In linear programming, the objective function is a mathematical expression that defines the goal of the optimization problem, typically to maximize or minimize some quantity. In the context of the provided question, where the focus is determining the optimal value of the objective function, it is critical to analyze the constraints and the feasible region defined by those constraints to identify the maximum or minimum value achievable.

To arrive at the optimal value of 6, the analysis likely involves inspecting the corner points of the feasible region formed by the constraints. When solving a linear programming problem, optimal solutions often occur at these corner points. The correct answer suggests that at a specific corner point of the feasible region defined by the problem's constraints, the value of the objective function reaches its maximum (or minimum) at 6, indicating that this point satisfies all constraints while providing the highest return according to the objective function.

In deriving the objective function value, one would typically substitute the coordinates of the corner points into the objective function to compute its value, ultimately determining that 6 is the highest value obtained among the feasible points. This reinforces how optimization in linear programming heavily relies on analyzing feasible solutions and their corresponding objective function outputs.