Final answer:
Pesticides are most ineffective and hardest to control during the pupae stage of a flea's life cycle, as the flea is encased in a cocoon and not active, which makes it resistant to pesticides.
Step-by-step explanation:
Within the flea's life cycle, pesticides are most ineffective and hardest to control during the pupae stage. The life cycle of a flea, like most insects, consists of several distinct stages: the egg, larva, pupa, and adult. It is during the pupa stage that the flea is encased within a cocoon and undergoing a process known as metamorphosis, where it transforms from its larval form into an adult. Pupae do not eat or move, making them particularly resistant to the effects of pesticides, which often target the more active and feeding stages of an insect's lifecycle such as the larvae and adults. This biological transformation allows the flea to remain protected from external influences, hence making the pupal stage the most challenging to target with pest control measures.