4. Given the function $f(x)=4 \sin x \cdot \sin ^{2}$ $\left(\frac{\pi}{4}+\frac{x}{2}\right)+\cos 2 x$, if the constant $w>0, y=f(w x)$ is an increasing function on the interval $\left[-\frac{\pi}{2}, \frac{2 \pi}{3}\right]$, then the range of $w$ is_ _. $\qquad$
4. Solution: $f(x)=2 \sin x+1$, and $f(w x)=2 \sin w x+1$

is increasing on $\left[-\frac{\pi}{2}, \frac{2 \pi}{3}\right]$, so $\left[-\frac{\pi}{2}, \frac{2 \pi}{3}\right] \subseteq$
$\left[\frac{2 k \pi-\frac{\pi}{2}}{w}, \frac{2 k \pi+\frac{\pi}{2}}{w}\right]$, where $k \in \mathbf{Z}$, hence $2 k-\frac{1}{2} \leqslant$ $\frac{-1}{2} w, 2 k+\frac{1}{2} \geqslant \frac{2}{3} w$, i.e., $4 k \leqslant 1-w-1$, thus $k=0$, so $w \in\left(0, \frac{3}{4}\right]$.