3. [7 marks] Let y1, ..., In be a set of responses and let $1, ..., 'n be a corresponding set of predictors lying in the interval [a, b]. Recall that a cubic smoothing spline is the function obtained by minimizing R(f) = _(y; - f(x; ) )2 + x / f" (x) da (1) i=1 over functions f in the space of all functions C defined on [a, b] for which fo f"(x)'dx is well defined (where > > 0 is a smoothing parameter). Write fx(x) for the minimizer. Consider the cubic smoothing spline f,'(x) obtained by minimizing [(; - f(z, ) )2 + x f"(2)dx (2) ifi over all f E C (that is, consider the smooth obtained when the ith point is deleted). a) Let y,, j = 1, ..., n, be the sequence with y; = y,, j # i, and y; = fx (xi). Show that the function minimizing [(y; - f(x; ) ) + f"(x)de (3) 1=1 over f E C is fx'(x). Hint: You can start by splitting the first term in expression (3) into a sum that includes all terms j / i and the ith term]