How to prove the following function is increasing function?

Jinliang

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May 12, 2016
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I am stuck on figuring out why the following function is a increasing function when I read a paper. The function is following


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


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.


attachment.php

The first derivative of f(x) is by the following:
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Note to viewers: Some other instances of this exercise are here, here, and here; and the exercise was "voluntarily" removed from here. ;)
 
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