Very interesting problem, nathanf534! It took me a very long time to find even one solution. A simple cubic won't work, even if you use as the independent variable. Neither will a straight-forward arctangent function work. I'm assuming you need as the independent variable. My solution is a pasting of two functions together at such that the function values and the function derivatives match up there. Is that an acceptable solution? I also have a vertical tangent at . I'm not sure if that's acceptable or not. Technically, the function is not differentiable there. I've tried making the solution better behaved there, but for some reason it's balking. Here is my solution so far:
Let me know if this works for you. I obtained this solution through a rather circuitous route. First I solved the same problem, but with The conditions turn out nicer and were easier to work with. I then transformed the solution into the proper domain by a linear transformation on . The ansatz was two quadratics in that I pieced together at the required point, matching up both function values and derivatives. I also forced the concavities in order to assure that the derivative was maximized at .