4.2.9 * Let $x$ be an acute angle, find the maximum value of the function $y=\sin x \sin 2x$.


$$
\begin{aligned}
y & =2 \sin ^{2} x \cos x \cdot y^{2}=4 \sin ^{4} x \cos ^{2} x=2\left(1-\cos ^{2} x\right)\left(1-\cos ^{2} x\right) \cdot 2 \cos ^{2} x \\
& \leqslant 2\left(\frac{\left(1-\cos ^{2} x\right)+\left(1-\cos ^{2} x\right)+2 \cos ^{2} x}{3}\right)^{3}=2\left(\frac{2}{3}\right)^{3}=\frac{16}{27},
\end{aligned}
$$

Therefore, $y \leqslant \frac{4 \sqrt{3}}{9}$, with equality when $\cos ^{2} x=\frac{1}{3}$.