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Let $y(t)$ be a smooth sigmoidal curve, $y^{(n)}(t)$ be its $n$th derivative,$\{t_{m,i}\}$ and $\{t_{a,i}\}$, $i=1,2,\dots$ be the set of points whererespectively the derivatives of odd and even order reach their extreme values.The "critical point of the sigmoidal curve" is defined to be the common limitof the sequences $\{t_{m,i}\}$ and $\{t_{a,i}\}$, provided that the limitexists. We prove that if $f(t)=\frac{dy}{dt}$ is an even function such that themagnitude of the analytic representation $|f_A(t)|=|f(t)+if_h(t)|$, where$f_h(t)$ is the Hilbert transform of $f(t)$, is monotone on $(0,\infty)$, thenthe point $t=0$ is the critical point in the sense above. For the general case,where $f(t)=\frac{dy}{dx}$ is not even, we prove that if $|f_A(t)|$ monotone on$(0,\infty)$ and if the phase of its Fourier transform $F(\omega)$ has a limitas $\omega\to \pm\infty$, then $t=0$ is still the critical point but as opposedto the previous case, the maximum of $f(t)$ is located away from $t=0$. Wecompute the Fourier transform of the generalized logistic growth functions andillustrate the notions above on these examples.

The Critical Point of a Sigmoidal Curve: the Generalized Logistic Equation Example