Question 120264
If the first mile is >= 45 min (3/4 hr.), the upstream trip will be >= 1 hr., leaving no time for the downstream trip.  The minimum upstream speed is therefore

(1 mi/0.75 hr) = 4/3 mph. (Minimum upstream speed)

The downstream speed cannot be less than the upstream speed, so the maximum downstream time is 30 min.  This would be equal to the upstream time of 30 min, allowing 15 min for the first mile, or 4 mph.

4 mph (Maximum downstream speed)

Let r be the speed of the boat and s be the speed of the stream.  Upstream speed is r-s, downstream is r+s.  From the above discussion,

r - s >= 4/3 and
r + s <= 4

or

r >= s + 4/3
r <= -s + 4

We also know that s >= 0.

Graphing, the regions overlap in a triangular region with vertices at (s,r) = 
(0,4/3)
(0,4)
(4/3, 8/3)

Possible Up- and downstream velocities are respectively
4/3, 4/3 (15 minutes upstream gives 1/3 mi)
4, 4  (15 min upstream gives 1 mi)
4/3, 4 (15 min upstream gives 1/3 mi)

Up- and downstream times are
3/4 + 1/4, 3/4 + 1/4 Total time = 2 hours.  Contradiction.
1/4 + 1/4, 1/4 + 1/4 Total time = 1 hour.  OK.
3/4 + 1/4, 4/3 + 1/4 Total time = 2 hrs. 35 min.  Contradiction.

The stream is not moving; the boat moves at 4 mph.

You can also simplify the problem initially--ignore the speed of the stream by assuming that it is zero.  In this case, the boat travels upstream and downstream at the same speed.  For the round trip to take 1 hr, it would be 0.5 hr upstream and 0.5 hr downstream.  The first mile would therefore take 15 min, the second part of the upstream trip would be 1 mi, and the entire trip would be 4 mi at a speed of 4 mph.  This works, and any other speed would result in a round trip either more or less than 1 hr.