2. [25 marks] Consider the following alternative representation for a cubic spline, S(r): a+b(x- 1)+c(x- 1)'+ }(x -1)'(x-2) 15:$2 S(I) = etf(x - 2) +g(x - 2)2 - 4(x -2)3(x -3) 251 53. We also wish our cubic spline to satisfy the boundary conditions: d's d's dz2 (1) = 0, dar2 (3) =0. (a) What are the conditions on the coefficients a, b, c, e, f and g such that S(x) interpolates the points (1,2), (2,1), and (3,1)? Deduce the values of a and e. (b) What is the condition on the coefficients such that S'(r) is continuous at r = 2? (c) Show that enforcing the boundary conditions at r = 1 and r = 3 leads to c = = and g = (d) Compute the values of b and f from part (a). (e) To ensure that S(r) is a cubic spline, what other condition needs to be checked? (It is not necessary to actually verify this condition for the purpose of this exercise.)