Is this function convex or concave?

Question

I have a function,

$f(x_{0},x_{1},......x_{n})=\sum^{n-1}_{i=0}A_{i}r^{-(x_{i}+x_{i+1})}-B$

$A_{i} >0$ for all $i$ and $B>0$ and $1 \leq r \leq 2$

so above function is convex or concave?

what would happen when $n=2$?

then, how to solve the problem below?

maximize $\sum^{n-1}_{i=0}A_{i}r^{-(x_{i}+x_{i+1})}-B$

such that $x_{i} > 0$ for all $i$

Answer 1

The answer is of course yes since your function is a sum of functions of the form $ r^{-x}$ (which is convex for $x>0$) composed with the linear function $x_i + x_{i+1}$. But convex composed with linear is convex, and that completes the proof.