6. Let $x$ be an acute angle. Then the maximum value of the function $y=\sin x \cdot \sin 2 x$ is . $\qquad$
6. $\frac{4 \sqrt{3}}{9}$.
$$
\begin{array}{l}
\text { Given } y=2 \sin ^{2} x \cdot \cos x \\
\Rightarrow y^{2}=4 \sin ^{4} x \cdot \cos ^{2} x \\
\quad \leqslant 2\left(\frac{2 \sin ^{2} x+2 \cos ^{2} x}{3}\right)^{3}=\frac{16}{27} \\
\Rightarrow y \leqslant \frac{4 \sqrt{3}}{9} .
\end{array}
$$

When $\cos ^{2} x=\frac{1}{3}$, the equality holds in the above expressions.