0
$\begingroup$

Consider a function f:[0,1] -> R with f(0)=0
having a finite derivative at each x in (0,1)
Prove that if f' is an increasing function (at least on the interval (0,1))
then h(x)=f(x)/x is also increasing

  • 0
    f(x) = -sin(x) is a counter example.2010-11-23

2 Answers 2

1

Hint: show that h'(x) is positive on $(0,1)$, using the mean value theorem applied to $f(x)-f(0)$.

  • 1
    Maybe I don't understand your comment. Can you please formulate it differently? (h'(x) > 0 for $x \in (0,1)$ for sure). Also note that MVT should be applied to $f$, not $h$.2010-11-23
0

This seems to be a straightforward application of Cauchy's Mean value theorem.

http://en.wikipedia.org/wiki/Cauchy%27s_mean_value_theorem#Cauchy.27s_mean_value_theorem