Q.In a Linear Programming Problem, the corner points of the feasible convex region are $(0, 0)$, $(0, 4)$, $(2, 3)$ and $(5, 0)$. Find the corner point at which the objective function $Z = 3x + 4y$ will be maximum.

Evaluate $Z=3x+4y$ at each corner point; the largest value, $18$, occurs at $(2,3)$.

Concept. For a linear programming problem, the optimal value of a linear objective function over a bounded convex feasible region always occurs at one of its corner (vertex) points — the corner-point theorem central to NCERT Class 12 mathematics.

Evaluate $Z=3x+4y$ at each corner:

$Z(0,0)=3(0)+4(0)=0,$

$Z(0,4)=3(0)+4(4)=16,$

$Z(2,3)=3(2)+4(3)=6+12=18,$ …