2) F(x) increases on the intervals where F'(x)>0. F'(x)=sin(x^{2}). Since 0≤x≤3, F'(x)≥0 on the intervals [0,√π)]∪[√(2π);3] Therefore, those are the intervals where F(x) increases.

3) Average rate of change of F(x) is simply this: (1/3)∫_{0}^{3} sin(t^{2})dt. It is k, by the problem statement. So, the answer here is simply 3k.