Example 1 Let $x, y \geqslant 0$, and $x^{3}+y^{3}=1$, find the range of $x+y$.


Using the result of Theorem 2, we get $x+y \leqslant\left(x^{3}+y^{3}\right)^{\frac{1}{3}}(1+1)^{\frac{2}{3}}=2^{\frac{2}{3}}$, and by $\left(x^{3}+y^{3}\right)^{\frac{1}{3}} \leqslant x+y$, we obtain $1 \leqslant x+y \leqslant 2^{\frac{2}{3}}$.