Linear Programming Problem

Step 1: Understand the problem.
We are given the objective function , and the constraints .
We need to find the maximum value of .
Step 2: Analyze the constraints.
The constraints define a feasible region on the coordinate plane.
The line intersects the x-axis at and the y-axis at .
Since and , the feasible region is a triangle with vertices at .
Step 3: Evaluate at the vertices of the feasible region.
At , .
At , .
At , .

Step 4: Conclusion. The maximum value of occurs at the point , where .
Therefore, the correct answer is option (d) 16, which is the maximum value among the given choices.

Step 1: Understanding the Theorem
This theorem refers to the fact that in optimization problems, the maximum or minimum of the objective function occurs at the vertices (corner points) of the feasible region.
This is called the fundamental theorem of linear programming and is associated with the concept of extreme points.
Step 2: Explanation of Other Options
Option (a) Algebra is not correct because algebra is not specifically associated with optimization problems involving extreme points.

Option (b) Arithmetic is not related to optimization theorems.

Option (c) Calculus involves the study of rates of change but does not directly address this theorem in the context of linear programming.

Step 3: Conclusion
Thus, the correct answer is Extreme Points.