$3\mathrm{.}1$ Two Days $(20$ pt$.)$

Suppose we are only looking at prices over two days $($i$.$e$.,$ n $=2).$ Design an O$($kP$)$ dynamic programming algorithm that computes the maximum amount of profit you can make buying stocks on the first day and selling stocks on the second day. Prove the runtime and correctness of your algorithm.

Hint: Let M$[$l$]$ be the maximum amount of profit you can make by buying l dollars of stock on the first day. Write a recursive relationship for M$[$l$].$