Why did Prospro give Ferdinand a task to try and prove to him that he can take care off Miranda?
Prospero's task for Ferdinand can be interpreted in several ways.
1. Prospero gives Ferdinand the task to test his love for Miranda. Ferdinand is the first "real" man Miranda has ever met; Prospero is playing the "protective father" role to ensure that Ferdinand's love is real and not fickle.
2. Ferdinand is a nobleman--King Alonso's son. It was the king who conspired with Antonio to have Prospero removed. It was the king's army who physically removed Prospero and Miranda and carried them to the island where they currently reside. Because of these factors, Prospero is getting revenge through Ferdinand.
3. "The Tempest" is a play about power and the effects of power. Prospero is proving how powerful he is by forcing Ferdinand to perform the task.