Consider the data points {(-1, 4), (0, 6), (1, 12)}. (a) Using the method of undetermined coefficients, derive the monomial basis form of the quadratic polynomial interpolating this data. Show all details of your work, including the Gauss transforms and permutation matrices used during the factorization of the Vandermonde matrix. (b) Derive the Lagrange form. Show all of your work. (c) Derive the divided-difference form. Again, show all of your work, including the divided- difference table. (d) Verify that the polynomials in (a), (b) and (c) are indeed the same polynomial by converting (b) and (c) to monomial basis form (i.e., write (b) and (c) as p(x) = 0 aix). (e) Append (2,16) to the data. Using the most efficient technique of the three for appending data points, construct the single polynomial that interpolates all four data points and convert it to monomial basis form. (f) Construct the linear spline interpolating all four data points {(-1, 4), (0, 6), (1, 12), (2, 16)}.