Question

In L.P.P., the corner points of the feasible region for the constraints \(3x-y\geq 6,x\leq 3,y\leq 2,y\geq 0,x\geq 0\) are .....

• A. \((3,2),(3,0),(2,0)\)
• B. \((\frac{8}{3},2),(3,2),(3,0),(2,0)\)
• C. \((0,0),(2,0),(\frac{8}{3},2),(0,2)\)
• D. \((3,2),(0,3),(0,2)\)

Hint:

Draw each constraint, find the boundary intersections that lie inside every other constraint, and list them.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The corner points of the feasible region are intersections of the boundary lines that satisfy all the constraints.

Step 2: Study the constraints.
\(3x - y \geq 6\), \(x \leq 3\), \(y \leq 2\), \(x \geq 0\), \(y \geq 0\). The line \(3x - y = 6\) meets \(y = 0\) at \(x = 2\), and meets \(y = 2\) at \(x = \dfrac{8}{3}\).

Step 3: Find the corners.
The region is the area to the right of the line \(3x - y = 6\), between \(y = 0\) and \(y = 2\), and to the left of \(x = 3\).
Corner 1: line with \(y = 0\): \((2, 0)\).
Corner 2: \(x = 3\) with \(y = 0\): \((3, 0)\).
Corner 3: \(x = 3\) with \(y = 2\): \((3, 2)\).
Corner 4: line with \(y = 2\): \(\left(\dfrac{8}{3}, 2\right)\).

Step 4: Check the options.
Option (B) lists exactly these four points. Option (A) misses \(\left(\dfrac{8}{3}, 2\right)\). Options (C) and (D) contain points such as \((0, 0)\) and \((0, 2)\), which fail \(3x - y \geq 6\).

Final Answer:
The corner points are \(\left(\dfrac{8}{3}, 2\right), (3, 2), (3, 0), (2, 0)\), option (B). \[ \boxed{\left(\frac{8}{3}, 2\right), (3, 2), (3, 0), (2, 0)} \]