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 $\displaystyle f(x)$ in the interval $\displaystyle [0, c]$, such that:

$\displaystyle f(0) = 0$

$\displaystyle f(b) = 1/2$

$\displaystyle f(c) = 1$,

where $\displaystyle 0 < b < c$.

This is a particular case of more general problem:

Find a real differentiable monotone function $\displaystyle f(x)$ in the interval $\displaystyle [a, c]$, such that:

$\displaystyle f(a) = A$

$\displaystyle f(b) = B$

$\displaystyle f(c) = C$,

where $\displaystyle a < b < c$ and $\displaystyle A < B < C$.

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, $\displaystyle f(x) = pe^{qx} + r$, where $\displaystyle p$, $\displaystyle q$ and $\displaystyle r$ are real parameters.

If applied to the more particular first case, I end up with the equation

$\displaystyle 2e^{bq} - e^{cq} - 1 = 0$.

I would be very grateful if you know such a function and share it.