# Bezier Curve and derivatives

Consider the following polynomial in Bezier form:

$$p(u)=\sum_{i=0}^n b_i B_i^n \Big(\frac{u-a}{b-a}\Big)$$

I am supposed to derive Bezier points $b_i$ for $i=0,1,...,n$ in general form, assuming that I know the values of the function, first derivatives and second derivatives at $a$ and $b$. I have tried using the recursive definition for the derivatives of Bezier polynomials but can't get anywhere.

You have six items of data (value, and first and second derivatives at each end). Therefore, you can calculate values for at most 6 of the control points $b_i$. So, what you're trying to do is possible only if $n \le 5$.
So, assuming $n = 5$, you just write down equations that express the various derivatives in terms of $b_0, \ldots, b_5$. So, for example \begin{align} p(a) &= b_0 \\ p(b) &= b_5 \\ p'(a) &= \frac{5}{b-a}(b_1 - b_0) \end{align} and so on. The left-hand sides are all known quantities, so you can solve these equations to get $b_0, \ldots, b_5$.