2. [25 marks] Consider the following alternative representation for a cubic spline, S(z): a+b(x - 1) + c(x - 1)? + 4(x -1)?(x -2) 15252 S(x) = etf(x - 2) +g(x - 2)2 - 4(x -2)2(x -3) 25153. We also wish our cubic spline to satisfy the boundary conditions: d's do2 (1) = 0, 2 2 (3) = 0. d's (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)