I'm looking for continuous function $f_k(x), \forall x\in[0,1]$ which are $C^N$ for $x\in(0,1)$ with given $N>0$, complying with this conditions:
$f_k(x)>0, \forall x\in(0,1)$
$f_k(0) = 0$
$f_k(1) = 1$
$f_k^{(i)}(0) = f_k^{(i)}(1) = 0, i=1,\dots,N$.
Finally, the goal is to construct these functions such that
$$ \lim_{k\to\infty} f_k(x) = 1, \forall x\in(0,1) $$
Inspired by the classical example of $x^{1/k}$, which complies $x^{1/k}\to 1$ as $k\to \infty, \forall x\in(0,1)$ I've been playing around with polynomials, fractional powers and combinations of these. However, I have only managed to comply the boundary conditions, but not succeed with the convergence part. I'm starting to suspect that this is not possible due to the flatness conditions at $x=0$ and $x=1$.
If its not possible, what is the closest we can get?