If the corner points of the bounded feasible region for a Linear Programming Problem (LPP) are A(0, 2), B(3, 0), C(2, 3) and D(3, 1), then the maximum value of the objective function $Z = 4x + 2y$ occurs at
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → every point on the line segment joining the points (2, 3) and (3, 1)
Given LPP with objective function: $Z = 4x + 2y$ and corner points of the bounded feasible region: A(0,2), B(3,0), C(2,3), D(3,1)
Compute $Z$ at each corner point:
A(0,2): $Z = 4*0 + 2*2 = 4$
B(3,0): $Z = 4*3 + 2*0 = 12$
C(2,3): $Z = 4*2 + 2*3 = 8 + 6 = 14$
D(3,1): $Z = 4*3 + 2*1 = 12 + 2 = 14$
The maximum value $Z_{\max} = 14$ occurs at points C(2,3) and D(3,1).
Check the line segment joining C and D using parametric form:
$x = 2 + t, \; y = 3 - 2t, \; 0 \le t \le 1$
Substitute into $Z$:
$Z = 4x + 2y = 4(2 + t) + 2(3 - 2t) = 8 + 4t + 6 - 4t = 14$
Since $Z = 14$ for all points on this line segment, the maximum occurs at every point on the line segment joining (2,3) and (3,1).