An Extreme Value Threshold

Let $f(x) = e^x - \dfrac{a}{x}$, where $a$ is a real constant.

1. For $a > 0$, find the intervals of monotonicity of $f(x)$. 2. If $f(x)$ has a critical point in $(0, +\infty)$ and the corresponding extreme value is greater than $\ln 4 + 2$, find the range of possible values of $a$.