I'm looking for the equation of a smooth curve that can describe the following behaviour:
$\bullet ~~f(0) = 1$
$\bullet ~~1 > f(i) > f(i + 1) > 0$ for all $1 \leq i \leq 4$
$\bullet ~~f(4) < f(5) < 1$
$\bullet ~~f(i) = 1$ for all $i \geq 6$
Only $f(0)$ and $f(6)$ are known with certainty, others are trial and error. Any suggestions how to generate a family of curves in this scenario with various curvatures? Thanks
I think you'll have to define it piecewise. For $i<6$, you can use something of the form $$\frac{Ai^2+Bi+1}{Ai^2+C}$$ and need to choose $A,B,C$ in order for the it to have the required behaviour. This function has an horizontal asymptote at 1 and a local maximum (and minimum if $B<0$) for values of $i$ close to zero. If there exist $A,B,C$ such that the derivative approaches zero as $i\rightarrow 6$ (I haven't checked) and you define the function to be $f(i)=1\quad \forall i\geq6$ it could be called smooth.