If $A = (a_1, \dotsc, a_n) \in \mathbb{R}^n$ is a vector, we define the function $f_A: \mathbb{R} \mapsto \mathbb{R}$ by $$f_A(x)= \sum_{i=1}^n \log(1 + |x - a_i|)$$ I have shown that the minimum of $f_A$ is reached for $x \in \lbrace a_1, \dotsc, a_n \rbrace$. This result can be obtained easily by considering that the function is piecewise concave on the $(a_i, a_{i+1})$ segments (considering ordered $a_i$). The concavity can be shown for example by taking the second derivative on the segment (inside a segment, the sign of $x - a_i$ is constant). However, it is not necessarily reached at the median value.
Is there any way to find analytically the value of the argmin of such function? Thank you!