Finding a monotone function crossing three given points

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.