1963 AHSME Problem 28

Given the equation 3x^2 - 4x + k = 0 with real roots. The value of k for which the product of the roots of the equation is a maximum is:

\textbf{(A)}\ \frac{16}{9}\qquad \textbf{(B)}\ \frac{16}{3}\qquad \textbf{(C)}\ \frac{4}{9}\qquad \textbf{(D)}\ \frac{4}{3}\qquad \textbf{(E)}\ -\frac{4}{3}

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