$f$ is continuous on $[a,b]$, differentiable on $(a,b)$ , why does that imply that $g(x)=\frac {f(x)} x$?

Question

Let $f$ be continuous on $[a,b]$, differentiable on $(a,b)$, $0<a<b$ and $\frac {f(a)} a= \frac {f(b)}b$.

Why does that imply we can define a function $g(x)=\frac {f(x)} x$ and what are the uses of this function?

(The context of this question: show that there exists $c\in (a,b) s.t: cf'(c)-f(c)=0$, I already know how to solve this with the above $g$)

Answer 1

When $0<a<b$ and $f$ is a real-valued function defined on $[a,b]$ then we are certainly allowed to define the function $$g(x):={f(x)\over x}\qquad(a\leq x\leq b)\ .$$ I cannot guess the "uses" of this function your author has in mind, but they will become apparent when you read on in your text.