Find $\alpha \in \mathbb{R}$ s.t. the second derivative ($x=0$) of the following function exists.

$f(x) =
\begin{cases}
e^{-\frac{1}{x}}, & \text{if $x$}\gt 0 \\[2ex]
\sin(x)+\alpha x^2-\log(1+x), & \text{if $x$ } \leq0
\end{cases}$

How to deal with this kind of exercise? Is it enough to use Taylor at second order and then looking for the differentiability of the function?