Q.The optimal value of the objective function of an LPP (when it exists on the feasible region) always occurs: (A) at the centre of the feasible region (B) at a corner point of the feasible region (C) at the origin (D) on the objective function's $y$-intercept

Evaluating $z = 3x + 2y$ at the corner points $(0,10)$, $(2,2)$ and $(4,0)$ gives $20$, $10$ and $12$; the least is $10$ at $(2,2)$.

By the corner-point (vertex) method, the optimum of a linear objective over a feasible polygon occurs at one of its corner points. Compute $z = 3x + 2y$ at each:

$z(0,10) = 3(0) + 2(10) = 20$

$z(2,2) = 3(2) + 2(2) = 6 + 4 = 10$

$z(4,0) = 3(4) + 2(0) = 12$

The smallest value is $z = 10$, attained at $(2,2)$.

The optimum value of an LPP's objective function occurs at a corner point (vertex) of the feasible region, not at its centre. So the statement is False.

The objective function $Z=ax+by$ is linear, and the feasible region of an LPP is a convex polygon (bounded case). The corner-point theorem states that if an optimal solution exists, then the objective function attains its optimum value at one of the vertices (corner points) of the feasible region. The reason is that as $Z$ increases or decreases, its level lines sweep across the region and last touch it at a corner, never settling at an interior point such as the centre.

Therefore the claim that the optimum occurs at the centre of the feasible region is incorrect.

Evaluate $z = 4x + 5y$ at the four corner points; the values are $0,\ 12,\ 13,\ \frac{35}{3}$, so the maximum is $13$ at $(2,1)$. Hence option (B).

The corner-point (vertex) method states that for a linear objective function on a bounded feasible region, the optimum occurs at one of the corner points. Evaluating $z = 4x + 5y$:

The largest value is $13$, occurring at $(2, 1)$.