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.
The 'proof' You have shown is really complicate and pratically impossible to undestand... very confortable and undestable is to consider that is...
(1)
... and from (1) You derive...
(2)
The first term of (2) is <0, the second >0 and because is for all k is ...
Kind regards
All right!... You have to consider the expression...
(1)
Of course for y=0 is...
(2)
... and the inequality...
(3)
... is reduced to an identity. The (3) is true if the second term is increasing with y so that we can compute...
(4)
... and it is easy to see that is for only if ...
Kind regards