Example 2 Find the minimum value of the function $y=\frac{2 \sqrt{2}}{\sin x}+\frac{1}{\cos x}, x \in$ $\left(0, \frac{\pi}{2}\right)$.
From the theorem, we get $y_{\min }=\left[(2 \sqrt{2})^{\frac{2}{3}}+1\right]^{\frac{3}{2}}$ $=3 \sqrt{3}$.