curve fitting with known first derivatives

Question

I need to find a polynomial function $f(x)$ minimum order 2 that best satisfies the following

it passes through points (x1,y1) and (x2,0)

it is known that $f'(x_{1})=A$ and $f'(x_{2})=B$ with $A<B<0$

and $A<f'(x)<B$ for $x_{1} < x < x_{2}$

it is monotonically decreasing

Any ideas? I started trying to solve for a third order polynomial as there were four conditions but ended up in a mess.

Jay

Answer 1

Third order sounds good! Note that $(x-x_2)$ is a factor and write down a general quadratic multiplied by this. Now solve for the three remaining coefficients. Not sure that there's a better way than this.