Example $2^{[2]}$ Let $0 \leqslant x \leqslant \frac{\pi}{2}$, find the range of $(\sin x)^{3}+(\cos x)^{3}$.


Solve: From
$$1=(\sin x)^{2}+(\cos x)^{2} \leqslant\left[(\sin x)^{3}+(\cos x)^{3}\right]^{\frac{2}{3}}(1+1)^{\frac{1}{3}},$$

we get

Then from this, we have
$$\begin{array}{c}
\frac{1}{\sqrt{2}} \leqslant(\sin x)^{3}+(\cos x)^{3} \\
{\left[(\sin x)^{3}+(\cos x)^{3}\right]^{\frac{1}{3}} \leqslant\left[(\sin x)^{2}+(\cos x)^{2}\right]^{\frac{1}{2}}=1,} \\
\frac{1}{\sqrt{2}} \leqslant(\sin x)^{3}+(\cos x)^{3} \leqslant 1
\end{array}$$