3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries.
\includegraphics[max width=\textwidth, alt={}, center]{372c062a-793f-4fb8-a769-957479f5fce7-05_846_833_365_614} The vertices of the feasible region are \(A ( 3.5,2 ) , B ( 1.5,3 ) , C ( 0.5,1.5 ) , D ( 1,0.5 )\).
The objective is to maximise \(P = x + 3 y\).

1. Find the coordinates of the optimum vertex and the corresponding value of \(P\).
2. Find the optimum point if \(x\) and \(y\) must both have integer values. The objective is changed to maximise \(P = x + k y\).
3. If \(k\) is positive, explain why the optimum point cannot be at \(C\) or \(D\).
4. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the optimum point.