The solution to the logistic differential equation is:
\[P(t) = \frac{P_0 Ke^{rt}}{(K-P_0) +P_0 e^{rt}} \]
To find the point of inflection, first find the second derivative:
\[\begin{align} P'(t) &= \frac{rP_0 K(K-P_0)e^{rt}}{((K-P_0) +P_0 e^{rt})^2} \\ P''(t) &= \frac{r^2 P_0 K (K-P_0) e^{rt}((K-P_0)-P_0 e^{rt})}{((K-P_0) +P_0 e^{rt})^3} \end{align}\]
The find the point of inflection is where \(P''(t) = 0\):
\[\begin{align} r^2 P_0 K (K-P_0) e^{rt}((K-P_0)-P_0 e^{rt}) &= 0 \\ (r^2) (P_0) (K) (K-P_0) (e^{rt})(K - P_0 - P_0 e^{rt}) &= 0 \end{align}\]
Let's look at each factor (shown in brackets). Since none of the factors in \((r^2) (P_0) (K) (K-P_0) (e^{rt})\) is 0. This must mean:
\[\begin{gathered} K - P_0 - P_0 e^{rt} = 0 \\ K - P_0 = P_0 e^{rt} \end{gathered}\]
Making \(t\) the subject:
\[\begin{gathered} \frac{K - P_0}{P_0} = e^{rt} \\ t = \frac{1}{r}\ln \left( \frac{K - P_0}{P_0} \right) \end{gathered}\]
If \(P_0>K\), then this quantity is undefined, and the graph does not have a point of inflection. In the logistic graph, the point of inflection can be seen as the point where the graph changes from concave up to concave down.