(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

3. 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.

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# Homework Help: Prove a composite function is increasing

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