I will just give the solution for 2. (a) since the remaining case is very similar.
Clearly, if is increasing then we know that for all .
Hence, we have
which completes the first part of the inequality.
For the remaining part, we consider that is concave down (convex).
This indicates that holds for all (draw a graph for the increasing convex function and the line passing at the points and ).
Integrating both sides of this inequality, we get
which completes the proof.