$C^\infty$ approximations of $f(r) = |r|^{m-1}r$ Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$.
Is it possible to find $C^\infty$ functions $f_n$, such that 


*

*$f_n \to f$ uniformly on compact subsets of $\mathbb{R}$,

*$f_n' \to f'$ uniformly on compact subsets of $\mathbb{R}$, 

*$f_n'(0) > 0$, 

*$f_n(0) = 0$


and do we get any upper/lower bounds on $f_n$ and $f_n'$, and if so, how do they depend on $n$?
It is possible to find $f_n \in C^\infty$ such that 1,3,4 hold. 
It is possible also to find $\tilde f_n \in C^\infty$ such that 1 and 2 hold.
But how about all four together?

What about
$$
f_n(r) = \left(\sqrt{ r^2 + n^{-1}}\right)^{m-1}r
$$
