find a formula for the function

Question

y=a/1+be^-t with a y intercept at y=2 and an inflection point at t=1

Answer 1

another problem that we can solve with an understanding of distributions. This is a multiple of the cdf of a logistic function. So that the function's inflection is at t = 1, b = e (this makes be^-1 = 1; the logistic function's inflection at y=1/1+1 = 1/2)

Thus, at t = 0, we have, y = a/1+b*1 = a/1+e

y = 2 at 0, so a/1+e = 2, or a = 2 + 2e

Using Calculus, f'(t) = abe^-t/(1+be^-t)^2

f''(t) = abe^-t(-(1+be^-t)+2be^-t)/(1+be^-t)^3 =

abe^-t(be^-t-1)/(1+be^-t)^3

be^-t = 1

This is exactly what we got above; the inflection is at t = 1, so be^-t = be^-1 = 1, so b = e

Working similarly at x = 0, we get a = 2 + 2e.