The corner points of the feasible region of a LPP with the constraint…

The correct answer is Option (4) → \((15, 0), (40,0), (4, 18), (6,12)\)

The corner points are the intersections of the boundary lines that satisfy every constraint.

\(x + 2y = 40\) and \(3x + y = 30\): from the second, \(y = 30 - 3x\), so

    \(x + 60 - 6x = 40,\quad x = 4,\quad y = 18\)

    Check: \(4(4) + 3(18) = 70 \ge 60\), so (4, 18) is a corner point.

\(3x + y = 30\) and \(4x + 3y = 60\): with \(y = 30 - 3x\),

    \(4x + 90 - 9x = 60,\quad x = 6,\quad y = 12\)

    Check: \(6 + 2(12) = 30 \le 40\), so (6, 12) is a corner point.

\(x + 2y = 40\) and \(4x + 3y = 60\) meet at (0, 20), but there \(3x + y = 20 \lt 30\), so it is not feasible.

On the x-axis (y = 0) the constraints give

    \(x \le 40,\quad x \ge 10,\quad x \ge 15\)

so the region meets it from (15, 0) to (40, 0). On the y-axis (x = 0) they need \(y \le 20\) and \(y \ge 30\), which is impossible.

So the corner points are (15, 0), (40, 0), (4, 18) and (6, 12).

Shortcut: Test one point of each wrong option against the constraints:

    (0, 0): \(3x + y = 0 \lt 30\), so Option (1) is out
(0, 30): \(x + 2y = 60 \gt 40\), so Option (2) is out
(0, 20): \(3x + y = 20 \lt 30\), so Option (3) is out