Find both lower and upper asymptotic bounds for $T(n) = 2T(\frac{n}{2})+n^4$
It seems that you have overlooked the fact that $f(n) \in \Theta(n^4)$ already implies both an upper bound of $f(n)\in O(n^4)$ and a lower bound of $f(n)\in \Omega(n^4)$. Intuitively, $\Theta$- notation says that a function grows "as fast as" another function, which means both "at most as fast" and "at least as fast".