Given $f\in C^{\infty}(\Omega)$, with $\Omega \subseteq [-1,1]^2$, define
$$M[f](x):=\max_{y:(x,y)\in \Omega} f(x,y).$$
It is easy to show that this function may not be smooth: if $f(x,y)=xy$ and $\Omega=[-1,1]^2$ we have
$$M[f](x)=|x|\notin C^{1}(\Omega).$$
Eevn for $\Omega = B_1(0)$, the unit ball, which has smooth boundary, we get
$$M[f](x)=|x|\sqrt{1-x^2}\notin C^{1}(\Omega).$$
which is not smooth either. Note that that by Danskin's theorem we have a sufficient condition for $M[f]\in C^1$, but I know no results for higher order of differentiability. Is there some assumption on $f,\Omega$ under which $M[f]$ is more than one time differentiable?