Question

If the constraints of a Linear Programming Problem are : \(x + y \leq 6, 2x + y \leq 8, x \geq 0, y \geq 0\), then a corner point of the feasible region is

• A. (6,0)
• B. (0,8)
• C. (2,4)
• D. (4,2)

Hint:

To quickly find the valid intercepts, choose the smaller intercept on each axis. For x: \(\min(6, 4) = 4 \implies (4,0)\). For y: \(\min(6, 8) = 6 \implies (0,6)\). This saves time on checking constraints.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Corner points (or extreme points) of a feasible region in LPP are the intersections of the boundary lines formed by the constraints.
The region is bounded by \(x=0, y=0, x+y=6\), and \(2x+y=8\).

Step 2: Key Formula or Approach:
1. Find the intersection of the two main lines.
2. Check the intercepts on the axes.
3. Ensure the points satisfy all constraints.

Step 3: Detailed Explanation:
1. Intersection of \(L_1: x + y = 6\) and \(L_2: 2x + y = 8\):
Subtract \(L_1\) from \(L_2\):
\[ (2x + y) - (x + y) = 8 - 6 \implies x = 2 \]
Substitute \(x=2\) into \(L_1\):
\[ 2 + y = 6 \implies y = 4 \]
So, (2, 4) is a potential corner point. Let's check it against all constraints:
\(2+4 = 6 \leq 6\) (True), \(2(2)+4 = 8 \leq 8\) (True), \(2 \geq 0, 4 \geq 0\) (True).
2. Intercepts:
For \(L_1\): \((6,0)\) and \((0,6)\).
For \(L_2\): \((4,0)\) and \((0,8)\).
Check \((6,0)\) in \(L_2\): \(2(6)+0 = 12 \not\leq 8\). (Invalid)
Check \((0,8)\) in \(L_1\): \(0+8 = 8 \not\leq 6\). (Invalid)
The valid intercept corner points are \((4,0)\) and \((0,6)\).
The set of corner points is \(\{(0,0), (4,0), (2,4), (0,6)\}\).

Step 4: Final Answer:
Among the given options, (2, 4) is a corner point of the feasible region.