Example 1 Try to compare $\sqrt[3]{3-\sqrt[3]{3}}+\sqrt[3]{3+\sqrt[3]{3}}$ and $2 \sqrt[3]{3}$.
Analyze the function $y=\sqrt[3]{x}$ as a convex function.
Since $y=\sqrt[3]{x}$ is a convex function, by Jensen's inequality we have:
$$
\frac{\sqrt[3]{3-\sqrt[3]{3}}+\sqrt[3]{3+\sqrt[3]{3}}}{2} \leqslant \sqrt[3]{\frac{(3-\sqrt[3]{3})+(3+\sqrt[3]{3})}{2}}=\sqrt[3]{3} ;
$$

Also, since $3-\sqrt[3]{3} \neq 3+\sqrt[3]{3}$, we have $\frac{\sqrt[3]{3-\sqrt[3]{3}}+\sqrt[3]{3+\sqrt[3]{3}}}{2}<\sqrt[3]{3}$.
Thus, $\sqrt[3]{3-\sqrt[3]{3}}+\sqrt[3]{3+\sqrt[3]{3}}<2 \sqrt[3]{3}$.