How to prove the following function is increasing function?

[Jinliang]

I am stuck on figuring out why the following function is a increasing function when I read a paper. The function is following

where where , and the independent variable x is on interval .

I have plotted the function curves of f(x) with different c by the following, and found that it seems to be monotonic function. However, it is too hard to me to prove it.

The first derivative of f(x) is by the following:

The second derivative of f(x) is by the following:

[elfmotat]

An increasing function is a function where [math]f'(x)>0[/math] for all [math]x[/math]. So you need to find a way to prove that the derivative of the function is always greater than zero.

 

(Hint: this will be easier to see if you combine those fraction terms into a single fraction.)

Edited May 20, 2016 by elfmotat