Doubts in Problem C codeforces( educational round 66)

Can anyone explain me problem C ELECTRIFICATION from Codeforces Educational Round 66 (Rated for Div. 2). My doubts are from the tutorial are: 1. Since we need the dk+1 distance that means we should minimize distance to k+1 points so why we are looking over k points .
2. How they got the expression for f(x).
The problem statement is
and tutorial link is

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