Prove a composite function is increasing

[Salzburg]

Homework Statement

Hi,
I have trouble proving this claim and would really appreciate your help =). Thank you in advance!
So here's the question: Suppose that f is a continuous function for all x>= 0 and differentiable for all x> 0. Also, f(0) = 0 and f' (1st derivative of f) is increasing on its domain Define
g(x) = f(x)/ x for x >0.
Show that g is increasing for all x >0.

The Attempt at a Solution

I used the Quotion Rule to write g' = [xf'(x) - f(x)]/ x^2.
For g to be increasing, g' must be >= 0 --> the question becomes: proving xf'(x) - f(x) >= 0.
Since f' is increasing -> f'(x) > = f(0) with x >0
--> It is sufficient to have xf'(0) - f(x) >0
I wrote f'(0): by definition of derivative:
f'(0) = lim [f(x) - f(0)]/ x-0 for x --> 0
= lim f(x)/x
Here I got stuck with the limit; it does not allow me to cross-multiple the terms.

 

[lanedance]

so rearranging it remains to show that
$$f'(x) > \frac{f(x)}{x}$$

continuous & differentiable work with the mean value theorem, so there exists 0<c<x such that
$$f(c) = \frac{f(x) -f(0)}{x-0}$$

 

[Salzburg]

Thanks for answering my question =). I couldn't believe that I didn't think about the Mean Value Theorem...