Your conditions do not single out a particular function. For instance, $(x^2-1)^2, \frac{\cos(\pi x)+1)}2$ work. The first is a polynomial so that could be nice. The second is trigonometric.
If you want one that extends by 0 to a $C^\infty(\mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $\exp(-1/(1-x^2))$.
Lets say $\mathcal F$ is the collection of functions $:[0,1]\to \mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:
* $\mathcal F$ is convex: $f,g\in\mathcal F$ and $\lambda\in[0,1]$ implies $\lambda f + (1-\lambda) g \in \mathcal F$.
* if $f\in \mathcal F$, and $s \ge 1$, then $f^s\in\mathcal F$.
Here's some graphs on Desmos:
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