The corner points of the feasible region for an LPP are: (0, 2), (5, 3), (6, 8) and (3, 0). Let the objective function be $Z=4 x+6 y$. The minimum value of Z occurs at:

any point on the line segment joining (3, 0) and (0, 2)

The correct answer is Option (4) → any point on the line segment joining (3, 0) and (0, 2)

$Z = 4x + 6y$

$Z(0,2)=12$

$Z(3,0)=12$

$Z(5,3)=20+18=38,\;\; Z(6,8)=24+48=72$

$\text{Minimum value} = 12 \text{ at } (0,2) \text{ and } (3,0)$

$\text{Since Z has same value at both vertices, it is constant along the line segment joining them}$

The minimum occurs at any point on the line segment joining $(3,0)$ and $(0,2)$.