In LPP maximize \( z=600x+400y \) subjected to constraints \( x+2y\le 12,\ 2x+y\le 12,\ 4x+5y\ge 20,\ x,y\ge 0 \). the corner points of the feasible region are :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (5, 0), (6, 0), (4, 4), (0, 6), (0, 4)
The boundary lines meet the axes at:
\( \displaystyle x+2y=12:\ (12,0),\ (0,6) \)
\( \displaystyle 2x+y=12:\ (6,0),\ (0,12) \)
\( \displaystyle 4x+5y=20:\ (5,0),\ (0,4) \)
Solving \( x+2y=12 \) and \( 2x+y=12 \): subtracting gives x = y, so 3x = 12 and the lines meet at (4, 4).
The feasible region lies in the first quadrant, on or below both lines x + 2y = 12 and 2x + y = 12, and on or above 4x + 5y = 20 (the origin does not satisfy \( 4x+5y\ge 20 \)).
Going round its boundary:
\( \displaystyle \text{on the x-axis from}\ (5,0)\ \text{to}\ (6,0) \)
\( \displaystyle \text{along}\ 2x+y=12\ \text{to}\ (4,4) \)
\( \displaystyle \text{along}\ x+2y=12\ \text{to}\ (0,6) \)
\( \displaystyle \text{down the y-axis to}\ (0,4) \)
\( \displaystyle \text{along}\ 4x+5y=20\ \text{back to}\ (5,0) \)
So the corner points are (5, 0), (6, 0), (4, 4), (0, 6) and (0, 4).
Why the other options are incorrect
Option (1) (0, 0), (5, 0), (0, 4) — Incorrect. These are the corners of the region \( 4x+5y\le 20 \); the constraint is \( 4x+5y\ge 20 \), and (0, 0) does not satisfy it.
Option (3) (6, 0), (12, 0), (4, 4) — Incorrect. (12, 0) does not satisfy \( 2x+y\le 12 \), since 24 > 12.
Option (4) (0, 6), (4, 4), (0, 12) — Incorrect. (0, 12) does not satisfy \( x+2y\le 12 \), since 24 > 12.