This is the last question of Huazhong University of Science and Technology for two consecutive years. In fact, the coefficient of 9 can be raised to 4 at most. Here is only the proof of 9. You can ask if you are interested.

Continuing the above method, if the interval is changed to [0, 1/4] and [3/4, 1], the coefficient can be increased to 8, and the best result obtained by this method is 8.

As for how to get 4, 4 is the best coefficient, it needs further detailed discussion. The landlord may wish to think about it and ask again if he can't think of it. The more general conclusion of this problem mainly comes from Sobolev space.