smc

By Vieta's Formulas, the product of the roots is $\frac{k}{3}$. This value increases as $k$ increases. Also, the quadratic's roots are real, then the discriminant is greater than or equal to zero, so $$16-12k\ge 0$$ $$-12k\ge -16$$ $$k\le \frac{4}{3}$$ Thus, the $k$ value that maximizes the product of the roots is $\frac{4}{3}$, which is answer choice $\boxed{\frac{4}{3}}$.