Feasible Region & Optimum — practice questions

Find all corners of the region defined by $x + 2y \leq 40$, $3x + y \leq 60$, $x \geq 0$, $y \geq 0$, checking each against every constraint.

A region is defined by $4x + 3y \geq 24$, $x + y \geq 7$, $x \leq 8$, $x \geq 0$, $y \geq 0$. Find its corners, state whether it is bounded, and explain what that means for maximising and for minimising a positive objective function over it.

Explain why the feasible region of a linear programme is always convex, and prove that a linear objective function attains its maximum at a corner of a bounded feasible region.

A feasible region has corners $(0,0)$, $(20,0)$, $(16,12)$ and $(0,20)$. Maximise $P = 5x + 7y$ and state which corner is optimal.

For the region with corners $(0,10)$, $(0,80)$, $(20,60)$, $(40,20)$, $(40,10)$, maximise $P = 30x + 30y$. Explain what you find, and describe how the sliding-line method reveals it.

For the dairy model (maximise $P = 40x + 30y$ subject to $2x + y \leq 100$, $x + y \leq 80$, $x \leq 40$, $y \geq 10$, $x \geq 0$), determine the range of Halloumi profit per tray for which $(20, 60)$ remains optimal, and analyse what happens outside that range.