Solution:

Applying $f(3x) = 3f(x)$ six times gives $f(2001) = 3^6 f\left(\frac{2001}{729}\right),$ and $\frac{2001}{729}$ lies in $[1, 3],$ so $$f(2001) = 729\left(1 - \left|\tfrac{2001}{729} - 2\right|\right) = 729 - |2001 - 1458| = 729 - 543 = 186.$$

For $x \in [3^k, 3^{k+1}],$ we have $f(x) = 3^k f\left(\frac{x}{3^k}\right) = 3^k\left(1 - \left|\frac{x}{3^k} - 2\right|\right),$ a tent whose maximum value is $3^k.$ To achieve $186$ we need $3^k \ge 186,$ so $k \ge 5,$ and the smallest solutions lie in $[243, 729],$ where $f(x) = 243 - |x - 486|.$

Setting $243 - |x - 486| = 186$ gives $|x - 486| = 57,$ so $x = 429$ or $x = 543.$ The smallest $x$ is $429.$