1. ## gamma-function

gamma-function

g(t) = ymax * t^(alpha) * exp( alpha * (1-t))

With known ymax and alpha, what is the maximum slope of g(t) (with t>0)?

2. ## Re: gamma-function

To compute the slope, you differentiate the function with respect to :

To find the maximum slope, we differentiate again:

Assume that such that

In such case the sign of solely depends on the sign of

So is a local maximum. And you just need to compare this with the boundary

But as you have the constraint , so when you compare the endpoints and instead.

It is easy to see that , therefore we conclude that the maximum slope is at when

Otherwise when , we see that which means that the slope is unbounded, or more strictly speaking the maximum point/maximum slope does not exist.

3. ## The Following User Says Thank You to BGM For This Useful Post:

pongtep (05-23-2013)

4. ## Re: gamma-function

It’s so perfect. I do appreciate your kindness.

5. ## Re: gamma-function

What value does the maximum slope have?

6. ## Re: gamma-function

The maximum slope is just when

and does not exist when ()

7. ## Re: gamma-function

What about highest value (that can be possible) such as 0.5, 1, 1.5, 3, 10, ..... something like that

8. ## Re: gamma-function

Not sure what do you want. You just have to plug in all the values you have.

9. ## Re: gamma-function

highest value = 1

10. ## Re: gamma-function

Ok let say you have the difficulty to plug-in the formula and I do it for you:

Now you can further put the (exact) values of to obtain the numerical value which you desperately want.

11. ## Re: gamma-function

It is highest result (numerical value) that it can calculate from this formula. I not sure, perhaps it is infinity.

12. ## Re: gamma-function

Let say you treat as a variable instead of a constant and you want to maximize over . Then you can see increase without bound as increase without bound. It is the same for

13. ## Re: gamma-function

Yes, and g' will increase without bound, is it correct?

14. ## Re: gamma-function

Yes you see the functions etc all have a factor in front. So they must increase as increase (when they are positive).

15. ## The Following User Says Thank You to BGM For This Useful Post:

pongtep (05-27-2013)

16. ## Re: gamma-function

Thank you so much. If i have further questions would you mind if I asked you again?

17. ## Re: gamma-function

ymax = 3
alpha = 2

What is maximum slope?

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