Question

The feasible region of an LP problem in two variables is defined by $ X_1 \leq 4.5, \quad X_2 \leq 3.7, \quad \text{and} \quad X_1 + X_2 \leq 7.0 $ along with the non-negativity constraints. The number of corner feasibility points are:

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

In linear programming problems, the feasible region is a polygon, and corner points occur at the intersection of constraint boundaries. Always verify each candidate vertex satisfies all constraints.

Answer 1

The Correct Option is B

We are given the constraints: \[\begin{align*} X_1 &\leq 4.5 \\ X_2 &\leq 3.7 \\ X_1 + X_2 &\leq 7.0 \\ X_1, X_2 &\geq 0 \quad \text{(non-negativity)} \end{align*}\] These inequalities define a polygonal feasible region in the first quadrant. To find the number of corner feasibility points (vertices of the feasible region), we determine where the boundary lines intersect:
Option (A) Intersection of \( X_1 = 0 \) and \( X_2 = 0 \): \( (0, 0) \)
Option (B) Intersection of \( X_1 = 0 \) and \( X_2 = 3.7 \): \( (0, 3.7) \) 
Option (C) Intersection of \( X_2 = 0 \) and \( X_1 = 4.5 \): \( (4.5, 0) \) 
Option (D) Intersection of \( X_1 + X_2 = 7 \) and \( X_2 = 3.7 \): solve \( X_1 + 3.7 = 7 \Rightarrow X_1 = 3.3 \Rightarrow (3.3, 3.7) \) 
Option (E) Intersection of \( X_1 + X_2 = 7 \) and \( X_1 = 4.5 \): solve \( 4.5 + X_2 = 7 \Rightarrow X_2 = 2.5 \Rightarrow (4.5, 2.5) \) 
Now we check which of these points satisfy all constraints: - \( (0,0) \), \( (0,3.7) \), \( (3.3,3.7) \): 
satisfy all constraints. - \( (4.5,0) \), \( (4.5,2.5) \): violate \( X_1 + X_2 \leq 7 \) 
So, valid corner points = 3.