The easiest way is to check the packaging. Read the labels and just take a look at the box and compare them to picture online. It's easy to spot.
As for physical headphones themselves, the fake ones will have much worse built-quality and usually some "fake looking" logo.
No, the Beats by Dr. Dre headphones do not have a radio on it.
Beats by Dr. Dre headphones are made in China.
No, Beats by Dr. Dre headphones aren't and never were made by Apple. Apple sells Beats by Dr. Dre headphones but that is all. Monster used to manufacture Beats by Dr. Dre headphones but recently their five year contract ended and Beats by Dr. Dre is making their headphones independently.
The function of Beats by Dr. Dre headphones are to listen to music with them.
Yes, Beats by Dr. Dre headphones work with Blackberries,
Beats by Dr. Dre headphones are popular in Canada and the USA.
Beats by Dr. Dre is a line of headphones co-owned by Dr. Dre himself.
No, Beats by Dr. Dre Solo headphones do not require batteries.
Be extremely careful when looking for cheap Beats by Dr. Dre headphones. Most of the Beats by Dr. Dre headphones on the market are counterfeits. The only way to be sure that the Beats by Dr. Dre headphones you are buying are real is to buy from an authorized retailer. Visit the link provided below for a list of authorized Beats by Dr. Dre headphones retailers.
Beats by Dr. Dre is a line of headphones.
David Guetta's Beats by Dr. Dre headphones are called Beats by Dr. Dre Mixr headphones. I have provided a link to them below this answer.
Chinese.