Why cubic curves provide the minimum curvature interpolants? As described by Shirley in his computer graphics book,


  Cubic curves provide the minimum-curvature interpolants to a set of points. That is, if you have a set of n + 3 points and define the “smoothest” curve that passes through them (that is the curve that has the minimum curvature over its length), this curve can be represented as a piecewise cubic with n segments.


Could you please give some hints or references about its proof?
Splines confuse me which is one reason I asked somebody else to write that chapter.

But I like this explanation:
https://www.johndcook.com/blog/2009/02/06/the-smoothest-curve-through-a-set-of-points/