An interesting problem. I have a few comments:
1. is superfluous if Unless, as I suspect is the case, is a number you know ahead-of-time. Certainly, if on the interval in question, then is the distance traveled, whatever that is.
2. Depending on the maximum slopes of the velocity, a and c, and the required distance traveled, d, you may not be able to produce a function satisfying all those requirements. In particular, if the required distance traveled, d, is large compared with a and c, you will not be able to exhibit the required function.
3. My solution: From the origin, start rising at the maximum slope a. Backwards from the endpoint, descend at the maximum slope c. Then fiddle with what's in-between those two lines to make the area equal to d. You could try to make the area under the extreme lines equal to half of d, and account for the other half of d by various shenanigans that allow you to change things and still keep the same area.