Small proximity of important points of a function Let $a,b,c$ be coprime integers with c greater than b and a,  $a^2 + b^2 \gt c^2$ and consider the function $f(x) = a^x + b^x - c^x$. It is easy to verify that there exist $r$ and $s$ such that $f(r) \ge f(x)$ for all $x$ and $f(s) = 0$. Prove the inequality below (in other words, when $c$ is big $r$ and $s$ are very close to each other  or even f-graph appears vertically from its peak). 

$$ 0 &lt; s-r &lt; \frac{\ln(\ln c)}{\ln c}$$
An answer is,concisely,as follow:
An answer is,concisely,as follow: