Asymptotically, for the degree $n \rightarrow \infty$, is there an approximation of local extreme points of the Legendre polynomial $P_n(x)$? I am particularly interested in the first (either decreasing or increasing order) local extreme point.
I saw post gives a good approximation of local maxima, but not sure if we have an approximation of the local maxima point.
In terms of the Jacobi polynomials $$ P'_n (x) = \frac{{n + 1}}{2}P_{n - 1}^{(1,1)} (x). $$ Thus, we need to estimate the zeros of $P_{n - 1}^{(1,1)} (x)$ closest to $\pm 1$. With the notation of $(18.16.\mathrm{ii})$ in the DLMF, these zeros are $\pm \cos (\theta _{n - 1,1} )$. From $(18.16.8)$, we have $$ \theta _{n - 1,1} = \frac{{j_{1,1} }}{{n + 1/2}} + \mathcal{O}\!\left( {\frac{1}{{n^3 }}} \right) $$ as $n\to +\infty$. Here $j_{1,1}= 3.83170597020751\ldots$ is the first positive zero of the Bessel function $J_1$. Thus the local critical points closest to the endpoints $\pm 1$ are $$ \pm\! \left( {1 - \frac{1}{2}\left( {\frac{{j_{1,1} }}{{n + 1/2}}} \right)^2 + \mathcal{O}\!\left( {\frac{1}{{n^4 }}} \right)} \right). $$ With more work, it is possible to obtain higher terms in the asymptotics.
Hoping that I properly remember, for the $k$-th root of $P_n(x)$, in decreasing order, Francesco Tricomi gave the asymptotics $$x_{n,k}=\Bigg[1-\frac1{8n^2}+\frac1{8n^3}+O\left(\frac{1}{n^4}\right)\Bigg]\cos\left(\frac{4k-1}{2(2n+1)}\pi\right)$$
