$13 \cdot 21$ Given the equation $3 x^{2}-4 x+k=0$ has real roots, if the product of the roots of this equation is maximized, then the value of $k$ is
(A) $\frac{16}{9}$.
(B) $\frac{16}{3}$.
(C) $\frac{4}{9}$.
(D) $\frac{4}{3}$.
(E) $-\frac{4}{3}$.
(14th American High School Mathematics Examination, 1963)
[Solution] If one of the roots $x_{1}, x_{2}$ is non-positive, then their product is obviously not the maximum. For two positive numbers with a fixed sum, their product is maximized when the two numbers are equal.
Given $x_{1}+x_{2}=\frac{4}{3}$,
when $x_{1}=x_{2}=\frac{2}{3}$, $x_{1} x_{2}=\frac{4}{9}$ is the maximum.
Thus, $k=3 x_{1} x_{2}=\frac{4}{3}$.

Therefore, the answer is $(D)$.