Example 1 Given $x>0$, find the minimum value of $f(x)=2 x+\frac{1}{x^{2}}$.
Analyze $f(x)=2 x+\frac{1}{x^{2}}=x+x+\frac{1}{x^{2}} \geqslant 3 \sqrt[3]{x \cdot x \cdot \frac{1}{x^{2}}}=$ 3, when and only when $x=\frac{1}{x^{2}}$, i.e., $x=1$, $f(x)$ has the minimum value 3.