Question $3.$ The rod$-$cutting problem consists of a rod of n units long that can be cut into integer$-$length pieces. The sale price of a piece i units long is Pi for $i=1,$dots,$n.$ We want to find the maximum total sale price of the rod by apply dynamic programming to the rod$-$cutting problem. Let $F(k)$ be the maximum price for a given rod of length k $.$

$.$ Give the recurrence on $F(k)$ and its initial condition$($s$).$

$2.$ What are the time and space efficiencies of your algorithm?

Now, consider the following instance of the rod$-$cutting problem: a rod of length $n=5,$ and the following sale prices $P1=2,P2=3,P3=7,P4=2$ and $P5=5.$

$.$ Explain the execution of your dynamic programming algorithm $($in particular portray the computations of the different entries of the table$)$ and give its solution $($the total price and the actual cuts$).$