The correct answer to the question is option 4, Bal Gangadhar Tilak. He is considered the father of the extremist movement in India. The extremist movement was a radical approach to the fight for independence from British rule. Bal Gangadhar Tilak played a crucial role in shaping this movement and mobilizing people against the British regime.

Option 1, Motilal Nehru, was a prominent leader of the Indian National Congress and played a crucial role in the fight for independence, but he is not considered the father of the extremist movement.

Option 2, Gopal Krishna Gokhale, was a moderate leader and believed in using peaceful means to achieve independence. He is also not associated with the extremist movement.

Option 3, Vallabhbhai Patel, also known as the Iron Man of India, was a key figure in the struggle for independence, but he was not the father of the extremist movement.

In conclusion, Bal Gangadhar Tilak was the primary leader and initiator of the extremist movement in India.