2020 AMC 12B Problem 22

What is the maximum value of \frac{(2^t-3t)t}{4^t} for real values of t?

\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.