Find inflection points

Question

I am formalizing a stochastic and translating into the Stan probabilistic programming language.

Long story short: it's a measurement error model with a sub-model that uses the following function $y(t)$. But I need some additional information from that function and I'm asking your help with calculus -and maybe with Wolfram alpha for an approximate solution if there is'n a closed formulation- to find the inflection points (second order derivative in respect of $t$) of the following function (the panettone function): $$ y(t) = A \sin^n\left(\pi\,\frac{t-t_s}{t_r-t_s}\right) $$ that is defined in the domain $(t_r, t_s)$ there $0 < t_r< \frac{1}{2}$ and $\frac{1}{2} < t_s < 1$.

$A>0$, $n>0$, $t_s$, and $t_r$ are parameters. The function has a maximum for $t=\frac{1}{2}$.

I wrote on the paper the first derivative but I'm stuck writing the second derivative.

Thanks for any help!

Answer 1

Note that $y(t)$ has a maximum at $t = (t_r+t_s)/2$. The second derivative of $y(t)$ is $$ y''(t) = \frac{A n^2 \pi^2}{2(t_s-t_r)^2} \sin^{n-2}\left(\pi\frac{t_s-t}{t_s-t_r}\right)\left(\cos\left(2\pi \frac{t_s-t}{t_s-t_r}\right) + 1 - \frac{2}{n}\right). $$ The goal is to find any inflection points in the open interval $(t_r,t_s)$. There are none for $n \le 1$. For $n>1$ the inflection points are at $$ t = \frac{t_r+t_s}{2} \pm \frac{t_s-t_r}{2\pi} \cos^{-1} \left(\frac{n-2}{n}\right). $$