Here's my attempt:
f(x) measures the distance of x to 16 points, namely 2, 4, 8, 16, ... , 2^16.

Suppose we let x to be less than 2. Then we can decrease f(x) by increasing x towards 2 since this decreases the distance of x to all 16 points.

So what happens when we increase x pass 2? We're getting further away from a single point, ie 2, but at the same time, we are decreasing our distance to 15 points. The net effect is a decrease in f(x).

The key observation here is that if we change x by a fix amount, the change in the distance is the same for all points(some are positive change, while others are negative, but the absolute value is the same).

This implies that we should keep increasing x to decrease f(x) as long as we are decreasing the distance to more points than the number of points that we are getting further away.

Following this logic, we arrive at the answer that f(x) is minimum between the region 2^8 and 2^9, which has 8 points less than it and 8 points more.