The problem statement says to determine all node and branch currents. I've done so in my work attached, but that's not what I'm interested in.

What kind of circuit is this? How's it work? What change in the input signal (+5V on the left) would cause Q1 to turn off and Q2 to turn on?

 

Assume that the top transistor is just barely starting to turn on - so there is a tiny, tiny (i.e., basically zero) current flowing. What will the emitter voltage be? What will the base voltage be? What will the input voltage be? Now do the same assuming that the lower transistor is just barely starting to turn on.

 

What kind of circuit is this?

It's a push-pull output stage.