2 questions:

question 1: under what circumstances would this[O(f(n))=O(k.f(n))] be the most appropriate form of time-complexity analysis?

question 2: working from mathematical definition of O notation, show that O(f(n))=O(k.f(n)), for positive constant k.?

My view: For the first one I think it is average case and worst case form of time-complexity. am i right? and what else do i write in that?

for the second one I think we need to define the function mathematically, so is the answer something like because the multiplication by a constant just corresponds to a readjustment of value of the arbitrary constant 'k' in definition of O.