While solving some exercise I came up with this problem:
Assume that $f \in C^2[0,\infty]$ (i.e. $f$ has continuous second derivative on $[0,\infty)$ and there exists limit in $+\infty$ of $f$) be such that f''(0) = 0.
Does there exist a sequence $(f_n) \subset C^2[0,\infty]$ such that $f_n \to f$ and f''_n \to f'' in the supremum norm (i.e. $\lVert f_n - f \rVert_{C[0,\infty]}$, \lVert f''_n - f'' \rVert_{C[0,\infty]} \to 0 ) and all $f_n$'s satisfies f''_n(0) = f'_n(0) = 0?
Maybe I am wrong but I think it would be nearly enough to approximate in that sense linear (in some neighborhood of $0$) functions.
I will be grateful for any hints.