Good morning, I have been working with piecewise functions lately. I am trying to completethis piecewise function, in order to make it smooth, with both first and second derivative smooth as well (continuous). The function im working with is $$ f_\delta(\tau):= \begin{cases} \hspace{0.3 cm} 0 \hspace{0.5 cm} \text{ if } \hspace{0.3 cm} \tau \geq \delta\\ - \frac{1}{\tau^2} \hspace{0.3 cm} \text{ if } 0 < \tau \leq \frac{\delta}{2} \end{cases} \\ f_\delta'(\tau) \geq 0 \hspace{1.0 cm} \forall \tau \in (0, \delta]$$ this applies for all $$\delta > 0$$ Does anyone have a solution?.
Most importantly, is there a way to solve a more general problem like this one? (some constructive pattern in literature) Many thanks in advance.