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.