Suppose the function

*f *has the following four properties:

1. *f *is continuous for *x >=*0*;*

2.

*f'*(*x*) exists for *x > *0*;*

3.

*f*(0) = 0*;*

4.

*f'*is monotonically increasing.

Put

g

(*x*) =f(x)/x

;

(*x > *0)

and prove that *g *is monotonically increasing. To better understand the various hypotheses,

sketch the graph of a couple of typical functions *f *and interpret *g *geometrically.

I started by trying some various functions for f.

I tried f(x)=x,f(x)=x^1/2. Those are the two I looked at so far. I was trying to see if maybe there were better ones to sketch.

Now interpreting g geometrically has me a little confused. I guess that would mean just sketching g. Now if f(x)=x, then g is the graph of 1. That doesn't seem right for the g to be monotonically increasing. For the othr, we have g=x^1/2/x=x^-1/2. Again, I get the feeling that g is not monotonically increasing. So, maybe I should have used different functions for f?