Given the situation in Theorem 4, write
f(x) = r0p0 (x) + r1p1(x) + ∙ ∙ ∙ + rmpm (x)
Suppose that f(x) has at most k roots for any choice of the coefficients r0,r1,..., rm, not all zero.
(a) Show that MTM is invertible if at least k + 1 of the xi are distinct.
(b) If at least two of the xi are distinct, show that there is always a best approximation of the form r0 + r1ex.
(c) If at least three of the xi are distinct, show that there is always a best approximation of the form r0 + r1x + r2ex.