Equal Values, Forced Close

Let $f(x) = 2x - a\ln x + 4a$ with $a \in \mathbb{R}$.

1. Discuss the monotonicity of $f(x)$. 2. Let $g(x) = f(x) - \sin x$. If there exist $x_1, x_2 \in (0, +\infty)$ with $x_1 \neq x_2$ and $g(x_1) = g(x_2)$, prove that