Help with this question. Im not sure why I have been getting the answer wrong. Could you explain?

graph is of s(t) , so speeding up means s'(t) > 0 ====> [ 3 , 4 ] ; thus [ 0 , 3 ] for slowing down...but the phrase " speeding up " bothers me , in particular if a) is correct then this means concave down { while it should be concave up if the velocity is increasing }....again if a) is valid then b) has [1,2] followed by [ 0 1 ] U [ 2 , 4 ]...whereas I believe it should be the opposite

If it is speeding up, its acceleration is increasing - that is the second derivative, or the change in slope of the graph. On (a), the slope is decreasing from 0 to 2 and then starts increasing after that so I think you should put (2, 4] for the first answer and [0, 2) for the second.
On the next it looks like the slope is increasing from 0 to 1 then decreasing from 1 to 2 then increasing again from 2 to 4 so first answer is [0, 1) U (2, 4] and second is (1,2)

You might check out "convex up" and "convex down" for more on this topic.
IMO anyway