\begin{equation*} f''(x) = \frac{\sqrt{2} {\left(\mu + \sigma - x\right)} {\left(\mu - \sigma - x\right)} e^{\left(-\frac{\mu^{2}}{2 \, \sigma^{2}} + \frac{\mu x}{\sigma^{2}} - \frac{x^{2}}{2 \, \sigma^{2}}\right)}}{2 \, \sqrt{\pi} \sigma^{5}} \end{equation*}