$(20$ points$)$ Quick Sort is a sub$-$quadratic sorting algorithm that inputs a list of comparable objects, picks an element from the list as the "pivot", partitions the list into two separate sub$-$lists $($values less than pivot and values greater than pivot$),$ and finally does this procedure recursively for each sub$-$list. The following function implements quick sort in F#:

let rec qsort:int list $->$ int list $=$ function

$|[]->[]$

$|$ x::xs $->$ let smaller $=[$for a in xs do if a$<=$x then yield a$]$

let larger $=[$for b in xs do if b$>$x then yield b$]$

qsort smaller @ $[$x$]$ @ qsort larger

a$.(2$ points$)$ Identify the base case of this recursive function.

b$.(3$ points$)$ Which element of the input list is considered to be the pivot in each recursive call?

c$.(8$ points$)$ Write the input list of every recursive call to qsort function when we want to sort the list of integers $[5$; $3$; $7$; $1$; $10]$

d$.(7$ points$)$ Manipulate this function so that it sort a given list in decreasing order.