11. Since $f(x)=(3 x-1)\left(\sqrt{(3 x-1)^{2}+4}+1\right)+(2 x-3)\left(\sqrt{(2 x-3)^{2}+4}+1\right)$, let $g(t)=$ $t\left(\sqrt{t^{2}+4}+1\right)$. It is easy to see that $g(t)$ is an odd function, and $f(t)$ is a strictly increasing function. Therefore, $f(x)=g(3 x-1)+g(2 x-3)$. When $f(x)=0$, $g(3 x-1)=-g(2 x-3)=g(3-2 x)$, so $3 x-1=3-2 x$, solving for $x$ gives $x=\frac{4}{5}$. Hence, the coordinates of the intersection point with the $x$-axis are $\left(\frac{4}{5}, 0\right)$.
11. Find the coordinates of the intersection points of the graph of the function $f(x)=(3 x-1)\left(\sqrt{9 x^{2}-6 x+5}+1\right)+(2 x-3) \cdot\left(\sqrt{4 x^{2}-12 x+13}\right)+$ 1) with the $x$-axis.