Data structure with init $O(n)$ and interval max $O(\log n)$
Assume for simplicity that $n$ is a power of 2. Construct a complete binary tree in which the vertices at depth $k$ correspond to subarrays of length $n/2^k$, and contain as auxiliary information the maximum element of the subarray.

This data structure can be initialized in $O(n)$, and it supports your type of range queries in $O(\log n)$. You can also update the data structure by modifying arbitrarily a single element of the array in $O(\log n)$.

Details left to you.