It is well known that the four parameter logistic law has the following form $$ F(x)=D+\frac{A-D}{1+\Big(\frac{x}{C}\Big)^B} $$
What characterise this curve is its four parameters.
A=starting point of the curve.
B= steepness of the curve.
C=inflection point.
D= Maximum asysmptote.
If I am stating the definition of the parameters correctly, I am having a doubt about the parameter $C$. Indeed, if at $C$ this curve has an inflection point, then by definition of inflection point we must have $F''(C)=0$ if $F''$ exists. However, we have $$ F''(x)= −\frac{B(D−A)(\frac{x}{C})^B\Big((B+1)(\frac{x}{C})^B−B+1\Big)}{x^2\Big((\frac{x}{C})^B+1\Big)^3} $$ Thus $F''(C)= −\frac{B(D−A)}{4C^2} \neq 0$
So, what is $C$ ? Can $C $ be an inflection point ithout $F''$ being zero ?
Thanks in advance.
After correcting a mistake, I agree with the formula : $$ F''(x)= −\frac{B(D−A)(\frac{x}{C})^B\Big((B+1)(\frac{x}{C})^B−B+1\Big)}{x^2\Big((\frac{x}{C})^B+1\Big)^3} $$ Anyways $F''(C)\neq0$
This is obvious for low values of $B$. Moreover, there is no inflexion of the curve if $B\leq 1$ : in the formula below, $x_{inflexion}$ is not real.
In fact, saying that $C$ is the abscissa of the inflexion point is an approximate for loosely common use. The inflexion point is the more closer to $x=C$ the more the steepness $B$ is large.
The exact inflexion point is : $$x_{inflexion}=C \left(\frac{B-1}{B+1} \right)^{\frac{1}{B}}$$
The next graph shows in reduced coordinates the curves for various vales of $B$ and the real position of the inflexion points.
In fact $C$ is the value of $x$ for which $F(x)=\frac{A+D}{2}$ that is exactly the mean of $F(0)$ and $F(\infty)$.