Hi,
I have been trying from quite a while to solve the following problem.
Find a real differentiable monotone function in the interval , such that:
,
where .
This is a particular case of more general problem:
Find a real differentiable monotone function in the interval , such that:
,
where and .
I tried several different ways but ended up with some bad looking equations. My method was to define f as a function of some type with three parameters and try find these parameters from the above equations.
Example, , where , and are real parameters.
If applied to the more particular first case, I end up with the equation
.
I would be very grateful if you know such a function and share it.
Thanks for your consideration on my problem.
Today while thinking on it, I decided to apply the old Roman tactic, divide and conquer. So my plan was to find two functions. The first in the interval , crossing and . The second in the interval , crossing and . Also I needed those functions to have equal derivatives in point .
So, my first guess was two parabolas and , where . This is a solution of the problem at hand, but the characteristics of this function were not what I was looking for. I needed changes in around to yell greater changes in , than around the ends of the interval . So I defined and such that, and to be parabolas with the above requirements.
Another solution was to use ellipses. Taking the correct quarter of each ellipse where, the first one is with center and passes through and . The second is with center and passing through and . This defines a nice monotone differentiable function in .
EDIT: The function I found is not necessarily monotone, so sorry.
Try defining such a function:
Let , where A, B, C are constants we will work out.
I think this should be differentiable since its a polynomial function. All that remains is to restrict the function to the interval [0,c].
Well, here’s a modification of CaptainBlack’s suggestion.
Join the first two points ( and ) by a straight line. This will have gradient . Then join to by the function , where , , and .
Hopefully that might makes things a bit simpler.
By the way, CaptainBlack, the condition will make convex, and this will only work if lies below the straight line joining and . If lies above that line, then we need a concave function, which means we need both and to be negative ( to make the function concave and to make it increasing).
I expect my mental image of this was convex, so it works for what I was trying to do, its just that what I was trying to do was only half the problem!
In fact we need no restrictions on the constants, with the given relations between the values at the given points a function of the given form which passes through the data will be monotone increasing.
RonL