Describing the minimizers of this function

Question

Consider the following continuous function over $x$, with $a,b>0$: $$f(x)=\begin{cases} ax-\sqrt{x} & \text{for }x\leq b^{2}\\ ab\sqrt{x}-b & \text{for }x>b^{2} \end{cases}$$Note that this function is convex at first, then concave. Say I'd like to minimize $f(x_1)+\cdots+f(x_n)$, subject to the constraint that $x_1+\cdots+x_n = c>0$. What does the optimal solution look like as $c$ varies? Clearly, when $c$ is really small, the optimal solution has us setting all $x_i=c/n$ , and when $c$ is really large, we're going to have one value $x_i \approx c$ with all other values small. Can anything more precise be said?