We can improve the running time of quicksort in practice by employing insertion sort when the input is nearly sorted: Upon calling quicksort on a subarray with fewer than k elements, let it simply return without sorting the subarray. After the top-level call to quicksort returns, run insertion sort on the entire array to finish the sorting process.

(a) Argue that this sorting algorithm runs in O(nk + nlg(n/k)) expected time.

(b) How should we pick k? Discuss how this can be done both theoretically and practically.