How can I derive monotonically increasing polynomial functions with steep curves from [0,1] to [0,1]? For computer graphics reasons, I need a Taylor polynomial function on the interval [0,1] like the classic smoothstep function 3x^2-2x^3 which outputs a monotonically increasing value on the interval [0, 1] but that has a much steeper curve. It should have only one inflection point (not counting the beginning or the ends.) The beginning should be convex and the end should be concave.
I would also like to be able to tweak the function to my needs and make it steeper or not so steep with some parameters if possible.
How can I derive steep polynomial functions from [0,1] to [0,1]?

I want a polynomial, probably low-degree.
So let's try a + bx + cx^2 + dx^3 + e^4.
I want the point at zero to be zero so a must be zero.
I want the point at one to be one so c + d + e must be one.
I want the start to be convex and curve up so the first derivative at zero must be zero and hence b must be zero.
I want the end to be concave and curve down so the first derivative at one must be zero and hence 2c + 3d + 4e = 0.
Then d = 4 - 2 c and e = 1 - (c + d) and c is left free.
I am unsure of how to determine if c is two low or too high. I know that values lower than -1 don't work and values higher than around 7 don't work but I don't know why.
