I need a function (polynome) that values $0$ at $0$ and $1$ at $1$ and has these values as local maxima and minima. So far so easy the straight solution is:
$$f(x) = -x^4+2x^2$$
Now I want to parametrize the slope increase with which the function approaches $0$ and $1$ respectively (the pointedness or flatness of the maxima and minima). I further want the pointedness identical for $0$ and $1$. How can this be done?
Edit to make the purpose clearer:
I want a non linear function which is almost linear around 0.5 but then approaches 1(0) faster as x tends towards 1(0) respectively. The behavior should be the same for both edge cases.