# Increasing function

## Main Question or Discussion Point

Hi! I have the following question.
Let $f$ be continuous on $[0,\infty[$, $f(0)=0$, $f^\prime$ exists on $]0,\infty[$, and $f^\prime$ is increasing on $]0,\infty[$.

the question is to prove that the following function is increasing that is $g(x)=f(x)/x$ on $]0,\infty[$.

I tried to show that the first derivative is positive but I did not succeed to use the monotonicity of $]f^\prime[$

## Answers and Replies

If $f'$ is increasing, what does that say about the shape of $f$?

Can you say anything comparing $f'$ to $g$?

If the derivative is increasing that mean the original function is concave up, but how can that solve the question. I did not get it.

If the derivative is increasing that mean the original function is concave up...
Good first step.

2) Now, can you say anything about the comparison between $f'$ and $g$? Must one of them always be bigger than the other?

3) Is there a way to express $g'$ in terms of $f'$ and $g$?

I can't resist being... well, an economics nerd. This has a very clear meaning in public finance, for the case where $f(x)$ represents the number of tax dollars a person who makes income $x$ is required to pay. It says that if the marginal tax rate (i.e. how much you pay on your last dollar earned) is increasing, then taxation is progressive, i.e. higher income folks pay a higher overall tax percentage.

Office_Shredder
Staff Emeritus