The shaded region in the figure is the feasible region for x + y ≤ 4, x + 3y ≤ 6, x ≥ 0, y ≥ 0.

Explanation

1. 01

   The corner points of the feasible region are $(0, 0)$, $(0, 2)$, $(3, 1)$, $(4, 0)$.

2. 02

   Values of $Z$: $0$ at $(0, 0)$, $4$ at $(0, 2)$, $11$ at $(3, 1)$, $12$ at $(4, 0)$.

3. 03

   The region is bounded, so the maximum is the largest of these: $12$, at $(4, 0)$.