The minimum is attained at the mean
This is how i did it...
so whatever value minimises S would minimise
So if we look at all the values of x which minimises each term then the intersection of the values of x will minimise 2S and S.
looking at the graph of each term as chisigma did we see that all values between the minimum of each of the two terms are minimising values for both terms. since they are linear functions they intersect in the middle.
so the valuse minimising are and so on.
the intersection of
is so these values would all minimise S.
Problem: Known , minimize .
Two observations of are:
(1) It is uni-modal (for odd , 1 minimum point) or almost so (for even , 2 minimum points of equal functional value).
(2) The minimum value can be reached at the nodes .
With these observations, suppose minimum is reached at . Then
For the present example, , so or 9.