Decision tree complexity of finding parameters of "zigzag" array
The obvious lower bound is $\Omega(\log n)$, since there are $\binom{n+1}{2}$ possible answers. It is likely that using smart binary search you can find $i,j$ in $O(\log n)$ queries. I suggest starting with the easier case in which $j = n$, that is, the array increases and then decreases, and you want to find its maximum in $O(\log n)$ queries. Such arrays are known as bitonic, and indeed you can find $i$ in $O(\log n)$ comparisons (see for example this solution).