BIG OH ranking, Transitivity, and Code Snipet Runtime

Rank the following nine functions by order of growth, i.e., find an arrangement f_1, f_2, of the functions satisfying f_1 element O (f_2), f_2 element O (f_3), Partition your list into equivalence classes such that f and g are in the same class if and only if f element theta (g). For every two functions f_i, f_j that are adjacent in your ordering, prove shortly why f_i element O (f_j) holds. And if f and g are in the same class, prove that f element theta (g). 4^n, 3^n, 2_squareroot n, 2n^3 + 4, 1, log n, log log n, 3_squareroot n, 2_squareroot n^6 Bear in mind that in some cases it might be useful to show f (n) element o (g (n)), since o (g (n)) Subset O (g (n)). If you try to show that f (n) element o (g (n)), then it might be useful to apply the rule of 1'Hopital which states that lim_x rightarrow infinity f (n)/g (n) = lim_x rightarrow infinity f' (n)/g' (n) if the limits exist: where f' and g' are the derivatives of f and g, respectively. Use the definition of big-Oh to prove: If f (n) element O (g (n)) and g (n) element O (h (n)) then f (n) element O (h (n)) Give the theta-runtime for the code snippet below, depending on n. Justify your answer. for (i =n * n: i > = 1: i = i - 4) for (j = n * n: j > = 1: j = j/4) print (" ")