On square roots of class $C^m$ of nonnegative functions of one variable

We investigate the regularity of functions $g$ such that $g^{2}=f$,
where $f$ is a given nonnegative function of one variable. Assuming
that $f$ is of class $C^{2m}$ ($m > 1$) and vanishes together with its
derivatives up to order $2m-4$ at all its local minimum points, one can
find a $g$ of class $C^{m}$.
Under the same assumption on the minimum points, if $f$ is of class
$C^{2m+2}$ then $g$ can be chosen
such that it admits a derivative of order $m+1$ everywhere.
Counterexamples show that these results are sharp.