Solution

The correct answer is θ(n lg n).

This can be determined using the Master Theorem.

The Master Theorem is a general formula used to solve recurrence relations of the form T(n) = aT(n/b) + f(n), where a, b > 0 are constants, and f(n) is an asymptotically positive function.

In the given recurrence relation, a = 3, b = 4, and f(n) = n lg n.

Using the Master Theorem, we can see that the solution to the recurrence is T(n) = θ(n lg n), since log4 ³ = 1 and f(n) = n lg n = Θ(n lg n).

Therefore, the correct answer is θ(n lg n).