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.