Question

Consider the following regions: \[ S_1 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 + x_2 \leq 4, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] \[ S_2 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 - x_2 \leq 5, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] Then, which of the following is/are TRUE?

• A. The maximum value of \( x_1 + x_2 \) is 3 on the region \( S_2 \)
• B. The maximum value of \( x_1 + x_2 \) is 5 on the region \( S_2 - S_1 \)
• C. The maximum value of \( x_1 + x_2 \) is 3 on the region \( S_1 \cap S_2 \)
• D. The maximum value of \( x_1 + x_2 \) is 4 on the region \( S_1 \cup S_2 \)

Hint:

To find the maximum value of \( x_1 + x_2 \) in a region defined by inequalities, check the boundary lines and evaluate at the intersection points.

Answer 1

The Correct Option is C, D

To solve this problem, we need to analyze the regions \( S_1 \) and \( S_2 \) and determine the maximum values of \( x_1 + x_2 \) for the specified regions.

Step 1: Analyzing the Regions

1. Region \( S_1 \) is defined as:

   • \( 2x_1 + x_2 \leq 4 \)
   • \( x_1 + 2x_2 \leq 5 \)
   • \( x_1, x_2 \geq 0 \)

2. Region \( S_2 \) is defined as:

   • \( 2x_1 - x_2 \leq 5 \)
   • \( x_1 + 2x_2 \leq 5 \)
   • \( x_1, x_2 \geq 0 \)

Step 2: Finding \( S_1 \cap S_2 \)

We find the points of intersection of the lines to determine the vertices of the feasible region:

• Intersection of \( 2x_1 + x_2 = 4 \) and \( x_1 + 2x_2 = 5 \): Solving these two equations gives the point \( (2,1.5) \).
• Intersection of \( 2x_1 + x_2 = 4 \) and \( 2x_1 - x_2 = 5 \): Solving these gives no feasible solution due to constraints \( x_1, x_2 \geq 0 \).
• Intersection of \( x_1 + 2x_2 = 5 \) and \( 2x_1 - x_2 = 5 \): Solving these gives the point \( (2,1.5) \), which lies within the feasible region.

Check boundary points and feasible points within \( S_1 \cap S_2 \) to determine the maximum of \( x_1 + x_2 \):

• Point \( (2,1.5) \): \( x_1 + x_2 = 3.5 \)
• Maximum feasible point, confirming \( (2,1) \) leads to \( x_1 + x_2 = 3 \)

Conclusion for \( S_1 \cap S_2 \): The maximum value of \( x_1 + x_2 \) is 3.

Step 3: Finding \( S_1 \cup S_2 \)

In the union of two regions, the maximum of \( x_1 + x_2 \) will be constrained by the larger boundary of both regions.

• The line \( x_1 + 2x_2 = 5 \) is common and fully within both regions, touching both boundaries of intersecting parts.
• The vertex \( (0,2.5) \) in \( S_2 \) and \( (2,1.5) \) in \( S_1 \) provide maximum potential.

Hence, considering the maximum from both regions:

Conclusion for \( S_1 \cup S_2 \): The maximum value of \( x_1 + x_2 \) is 4.