The distance between $f$ and $g$ is $d = f-g$. Then the local maximum distance(s) between the two functions is/are at the solution(s) of $d'(x)=f'(x)-g'(x)=0$. i.e. $f'(x)=g'(x)$, so where they have equal gradients.
I am struggling to find a clear/convincing way to explain why this is true using the graphs and the gradients (rate of change).