I have a curve which passes through the origin with gradient zero, and also through the point (d,1) with gradient zero. It also passes through the point (m,n) between the two. I believe this is a quartic with an equation of the form y = ax^4 + bx^3 + cx^2. I need to be able to programmatically find the equation for different m, n, and d. So I need expressions for a, b, and c in terms of m, n, and d.
I have found these three equations:
(1) ad^4 + bd^3 + cd^2 = 1 (from point (d,1))
(2) 4ad^2 + 3bd + 2c = 0 (from gradient at (d,1), divided by d as I know d to be non-zero)
(3) am^4 + bm^3 + cm^2 = n
Can a, b, and c be found programmatically? If not is there another curve which fits my purposes (flat at origin and (d,1), passes through (m,n) somewhere between the two, I must be able to change d, m, and n)?
Many thanks in advance for the help.