Let $f, g : [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable at each $x \in (a, b).$

Question

Let $f, g : [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable at each $x \in (a, b).$ Prove that if $|f'(x)| \ge |g'(x)| > 0$ for all $x \in(a, b),$ then $ |g(x) − g(y)|\le |f(x) − f(y)|$ for all $x, y \in [a, b].$

My idea

since given

$f, g : [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable at each $x \in (a, b).$

By Cauchy's mean value theorem

then there exist at least one real number $ c \in (a,b)$ such that $\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)}$

so from this we can that result?

Answer 1

Almost there, just to write that $\left|\dfrac{f(x)-f(y)}{g(x)-g(y)}\right|=\left|\dfrac{f'(\xi_{x,y})}{g'(\xi_{x,y})}\right|\geq 1$ and we discuss only $x\ne y$.