Hello there - bit of an unusual question and I'm not quite sure if it belongs in this Calculus section or not. It sorta involves derivatives so here it goes:

I have been looking for monotonic equations that pass through (0,0) and (1,1) with various slopes at x=0 and x=1. For example, I already have:

CLASS 1,2: f(x)=x^n

when n>1: concave up, f'(0) = 0, f'(1) = n

when 0<n<1: concave down, f'(0) = inf, f'(1) = n

CLASS 3,4: f(x) = 1-(1-x)^n

when n>1: concave down, f'(0) = n, f'(1) = 0

when 0<n<1: concave up, f'(0) = n, f'(1) = inf

CLASS 5,6: f(x) = (1-(1-x)^n)^(1/n)

when n>1: concave down, f'(0) = inf, f'(1) = 0

when 0<n<1: concave up, f'(0) = 0, f'(1) = inf

What I am looking for are the equations that will have:

f'(0) = m

f'(1) = n

Thus, 2 different finite slopes at x=0 and x=1 allowing both concave up and down curves

Thank you for any help. I have asked math professors at my college but still no answer. Sorry for the long post...

Terry