11. Let $\mathrm{f}$ be a function defined on non-zero real numbers such that
$$
\frac{27 \mathrm{f}(-x)}{x}-x^{2} \mathrm{f}\left(\frac{1}{x}\right)=-2 x^{2},
$$
for all $x \neq 0$. Find $\mathrm{f}(3)$.
11. Answer: 2
Solution. Letting $x=-y$, we get
$$
-\frac{27 f(y)}{y}-y^{2} f\left(-\frac{1}{y}\right)=-2 y^{2} .
$$

Letting $x=\frac{1}{y}$, we get
$$
27 y f\left(-\frac{1}{y}\right)-\frac{1}{y^{2}} \mathrm{f}(y)=-\frac{2}{y^{2}} .
$$

Then $27 \times(1)+y \times(2)$ gives
$$
-\frac{729 \mathrm{f}(y)}{y}-\frac{\mathrm{f}(y)}{y}=-54 y^{2}-\frac{2}{y} .
$$

Solving for $\mathrm{f}(y)$, we have $\mathrm{f}(y)=\frac{1}{365}\left(27 y^{3}+1\right)$. Thus $\mathrm{f}(3)=\frac{3^{6}+1}{365}=2$.