Two Related Zeros

Let

where $e = 2.71828\ldots$ is the base of natural logarithms.

1. If $t = 0$, solve the inequality $f(x) + g(x) \geqslant 3$. 2. Let $t > 0$, and suppose $x_1, x_2$ satisfy $f(x_1) = 0$ and $g(x_2) = 0$.

1. Prove that $\dfrac{1}{e} < x_1 x_2 < e$.
2. If moreover $\ln\dfrac{x_1}{x_2} \geqslant 5 - 3\ln 2$, find the minimum value of $x_1 x_2$.