Minima of a function without derivation
Solution 1:
Note tha $\sqrt{a^2+x^2}$ is the distance from $p_1=(0,a)$ to $(x,0)$ and that $\sqrt{(b-x)^2 + c^2}$ is the distance from $p_2=(b,-c)$ to $(x,0)$. So all you meed to find is where the line crossing $p_1$ and $p_2$ is crossing the $x$ axis. This will be the point where your function achieves its minima.