Is there some rule about multiple functions(all continuous and differentiable in some specific interval) having their average slopes (as obtained using LMVT) be equal to the slope at the same point? As in, are the c-values of all functions in a specific interval the same?
Multiple problems I've come across seem to rely on this principle being true. If it is, how is it proved?
Consider $f(x)=\sin(x)$ and $g(x)=\cos(x)$ on $[0,2 \pi]$. You can actually find out the points where the slope is equal to the average slope and convince yourself that there is no common point for all.
There is one result in this direction, which looks like a generalisation: The Cauchy's mean value theorem. It says that if if the functions $f,g$ are continuous on $[a,b]$ and differentiable on $(a,b)$ then, there exists some $c \in (a,b)$ such that $$ f'(c) (g(b)-g(a))=g'(c)(f(b)-f(a)) $$
When, none of the terms above is zero, this relation can be written as $$ \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)} $$ which is similar but not the samea s what you want.