How to prove strict space lower bounds using crossing sequences in Turing machines?
A deterministic algorithm running in space $S$ runs in time $\exp S$. If you don't go over the entire array then you don't know the maximum number (formally speaking, consider two arrays differing in the last element, in only one of which it is the maximum). It takes time $n$ to go over the entire array. So $S = \Omega(\log n)$.