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?