11 The domain of the function $f(x)$ is $(0, \infty)$, and it satisfies $f(x)-2 x f\left(\frac{1}{x}\right)+$ $3 x^{2}=0$, then the minimum value of $f(x)$ is
113 Prompt: From $f(x)-2 x f\left(\frac{1}{x}\right)+3 x^{2}=0$, we can get
$$
f\left(\frac{1}{x}\right)-\frac{2}{x} f(x)+\frac{3}{x^{2}}=0 .
$$

By solving these two equations simultaneously, we get $f(x)=x^{2}+\frac{2}{x}$. Using the AM-GM inequality,
$$
\begin{aligned}
x^{2}+\frac{2}{x} & =x^{2}+\frac{1}{x}+\frac{1}{x} \\
& \geqslant 3\left(x^{2} \cdot \frac{1}{x} \cdot \frac{1}{x}\right)^{\frac{1}{3}} \\
& =3 .
\end{aligned}
$$

And the equality holds when $x=1$. Therefore, the minimum value of $f(x)$ is 3.