[Master of Complexity]: The Master Theorem might seem easy to work with, but in some cases, it may not apply directly. However, a good computer scientist can often find a creative way to make it work. Consider the following recursive relation:
$T(n) = 3T(n/2) + n^2\sqrt{n}\log\log n$
(a) Can the Master Theorem be applied directly to find the solution of the recursion? Justify your answer.
(b) We can express the recursion as $T(n) = 3T(n/2) + \Omega(n^\alpha)$. What is the largest value of $\alpha$? Justify your answer and use the master theorem to solve this recursion.
(c) We can also rewrite the recursion as $T(n) = 3T(n/2) + O(n^\beta)$. What are the possible choices for $\beta$? Justify your answer and use the master theorem to solve this recursion.
(d) Find a function $f(n)$ such that $T(n) = \theta(f(n))$ is the solution of the recursion.
