1. Let $0<x<4.5, f(x)=\left(1+\frac{1}{\lg \left(\sqrt{x^{2}+10}-x\right)}\right)\left(1+\frac{2}{\lg \left(\sqrt{x^{2}+10}-x\right)}\right)$, then the minimum value of $f(x)$ is $(\quad$.
A. $8 \sqrt{2}$
B. $8 \sqrt{3}$
C. $8+4 \sqrt{3}$
D. does not exist
1. D)

Let $a=\lg \left(\sqrt{x^{2}+10}+x\right), b=\lg \left(\sqrt{x^{2}+10}-x\right)$. Given $0 < x < 10$, we have $a > 0, b > 0$, and $a+b=\lg 10=1$.
$$
\text{Thus } \begin{aligned}
y & =\left(1+\frac{1}{a}\right)\left(1+\frac{2}{b}\right)=\left(2+\frac{b}{a}\right)\left(3+\frac{2 a}{b}\right) \\
& =8+\left(\frac{3 b}{a}+\frac{4 a}{b}\right) \geqslant 8+4 \sqrt{3},
\end{aligned}
$$

Equality holds if and only if $a=2 \sqrt{3} - 3, b=2(2-\sqrt{3})$, which implies $x=\frac{10^{2 a}-10}{2 \cdot 10 a}<0$, contradicting the given condition.