I know that the sequence \begin{equation*} f_n(x):=\begin{cases}\sqrt{x^2+\frac{1}{n^2}}-\frac{1}{n},& x\ge0, \\\\ 0,&\text{otherwise},\end{cases} \end{equation*} is a $C^1$ sequence pointwise convergent to $\max(0,x)$, with the further property $|f'_n(x)|\le M$ for all $n\in\mathbb{N}$, $x\in\mathbb{R}$ and for some $M\ge0$.
However, this sequence of functions turns out to be not twice differentiable and therefore not $C^2$. I tried to find a new sequence, but I was unsuccessful. So my question is: can someone provide a sequence of functions $(f_n)_n$ such that
- $(f_n)(x)\to\max(0,x)$ for all $x\in\mathbb{R}$;
- $f_n\in C^2(\mathbb{R})$;
- $f_n(0)=f_n'(0)=f_n''(0)=0$;
- $|f_n'(x)|\le M_1$, $|f_n''(x)|\le M_2$ for all $n$ and $x$ and for some $M_1,M_2\ge0$?