6、Let $x$ be an acute angle, then the maximum value of the function $y=\sin x \sin 2 x$ is $\qquad$

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6、Let $x$ be an acute angle, then the maximum value of the function $y=\sin x \sin 2 x$ is $\qquad$
Answer: $\frac{4 \sqrt{3}}{9}$.
Solution: Given $y=2 \sin ^{2} x \cos x$,
we get $y^{2}=4 \sin ^{4} x \cos ^{2} x=2\left(1-\cos ^{2} x\right)\left(1-\cos ^{2} x\right) \cdot 2 \cos ^{2} x$
$$
\leq 2\left(\frac{\left(1-\cos ^{2} x\right)+\left(1-\cos ^{2} x\right)+2 \cos ^{2} x}{3}\right)^{3}=2 \cdot\left(\frac{2}{3}\right)^{3}=\frac{16}{27} \text {, }
$$

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