Where exactly do you not understand the proof? What step doesn't make sense?
The problem is:
Show that the function is a monotonically decreasing function when , and , is a positive integer number.
A proof is as follows (but I cannot understand):
Consider the function , with . The first order derivative of with respect to is:
Because when , so , and finally, we obtain the function is monotonically decreasing function. The function is a special case of , and we have the proof for the problem.
The above proof is very short, and I cannot understand that. Can everyone give me some ideas?. Thanks in advance.
Thanks for your careful reading my problem.
+ First, how the first order derivative of is obtained?. I have implemented in Mathematica but the result cannot be simplified to that in the proof.
According to my knowlegde, when , the function is generalized from and can be written as:
. Is this right?. And if right, how to derive the .
+ Second, the way to show that when .
Thanks Ackbeet very much.
If K is an integer, the function can be expressed as:
. So the first order derivative of with respect to is:
And then, the problem turns into: prove that the when , , and .
And how this problem can be solved?. Thank you.
... and from (1) You derive...
The first term of (2) is <0, the second >0 and because is for all k is ...
Of course for y=0 is...
... and the inequality...
... is reduced to an identity. The (3) is true if the second term is increasing with y so that we can compute...
... and it is easy to see that is for only if ...