shortest path & max flow
What you're describing, where you minimize the sum of $f(u,v)w(u,v)$ with a given flow value $|f|$ isn't, in fact, the max-flow problem—it's the min-cost flow problem. If you constrain the flow value to be 1, and all capacities are set to 1, it's pretty clear that this problem is equivalent to finding the shortest path.

Basically, unit-capacity flow problems find edge-disjoint paths. If you set the flow value to be 1, you'll find only a single path. If you're working with min-cost flow, that path will necessarily be the shortest path (and, therefore, the cheapest flow).

If this is not what the book is asking, and it's really asking you about max-flow, I suspect there must be some details missing from your description. (For example, you could easily have a single node $t&#39;$ before $t$, with $c(t&#39;,t)=1$, so max-flow would have a flow value of 1, yielding a single path; however, that would still ignore the edge costs, and would just find any path, unless you're specifically using the Edmonds-Karp algorithm, which uses BFS to find its augmenting paths; in that case, you would, at least, find the shortest unweighted path.)