6. Let $x, y \in \mathbf{R}_{+}$. Then the function
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{x^{2}-x y+y^{2}}+\sqrt{x^{2}-9 x+27}+\sqrt{y^{2}-15 y+75}
\end{array}
$$

has the minimum value of
6. $7 \sqrt{3}$.

As shown in Figure 3.
Let $A B=3 \sqrt{3}, A C=5 \sqrt{3}, A M=x, A N=y$,
$$
\angle B A M=30^{\circ}, \angle M A N=60^{\circ}, \angle C A N=30^{\circ} \text {. }
$$

By the Law of Cosines, we get
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{x^{2}-x y+y^{2}}+\sqrt{x^{2}-9 x+27}+\sqrt{y^{2}-15 y+75} \\
=B M+M N+C N \\
\geqslant B C=\sqrt{27+75+45}=7 \sqrt{3} .
\end{array}
$$

When $x=\frac{45}{13}, y=\frac{45}{11}$, $f(x, y)$ achieves its minimum value $7 \sqrt{3}$.