The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$. 
Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval
$\textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad \textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad \textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad \textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad \textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$
Firstly note that $p \leq \frac{3}{4}$ and $p \leq \frac{2}{3}$, as clearly the probability that both $A$ and $B$ occur cannot be more than the probability that $A$ or $B$ alone occurs. The more restrictive condition is $p \leq \frac{2}{3}$, since $\frac{2}{3} < \frac{3}{4}$. 
Furthermore, by the [Inclusion-Exclusion Principle](https://artofproblemsolving.com/wiki/index.php/Inclusion-Exclusion_Principle), we also have \[\text{P}(A' \wedge B') = 1 - \text{P}(A) - \text{P}(B) + \text{P}(A \wedge B) = 1 - \frac{3}{4} - \frac{2}{3} + p = p - \frac{5}{12},\] and as a probability must be non-negative, $p - \frac{5}{12} \geq 0$, so $p \geq \frac{5}{12}$. Therefore, combining our inequalities gives $\frac{5}{12} \leq p \leq \frac{2}{3}$, or $\boxed{\textbf{(D)} \Big[\frac{5}{12},\frac{2}{3}\Big]}$.