Final stage resitor

[RapatchouX5]

My X5 has a defective AC. when I start the engine, the AC fails to deliver air from the front vents while the rear vents are working fine. I can hear the fan spinning but no air coming from the front vents, like something is obstructing the airflow (I feel air from the footwell vents and the rear vents)

Depending on my luck, the time needed to have AC cooling from the front upper body vents is ranging from 30 seconds to 5 minutes.

The AC compressor has been changed last year as well as the final stage resistor based on the advices from this web site. I have the followign questions:

1. How can I ensure that my final stage resistor has failed/ is broken and must be replaced?
2. What is the final stage resistor? What is the job of that part in the A/C system?
3. BMW is advising to change 8 fan motors for an amount of UD 2,500 but I refuse as my fan motors are working, they just need time to do their job.

Thanks in advance for your help