All parts of this question concern the function f(:c) : 6 sin :13 + 2 cos :5. (a) Find the smallest positive constant M that satisfies M 2 'f(k)(t)| for every possible combination of an integer k 2 0 and an evaluation point t e (00, +00). Hint: A standard trigonometric identity implies that, for a certain angle 45, one has f(:c) = V40 sin (a: -l- ) for all real as. Answer: M : sqrt 40 f(n+1) (15) 3:71" ( + 1)' for some 13 between 0 and a). This is n . Recall the standard decomposition f(:c) = Tum) + En(m), in which Lagrange's formula says En(:c) = valid for every integer n 2 0. In both parts below, estimate En(:c) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) (b) Find the smallest in, for which the polynomial value Tn(0.3) provides an approximation for f(0.3) that is guaranteed to be accurate to within 11 decimal places: Answer: n : Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |En(0.3)l S 0.5 X 107D. (c) Suppose n = 9 is prescribed. Find the largest positive number a such that the approximation Tg(:1;) for f($) is guaranteed to be accurate to within 9 decimal places, for all m in the symmetric interval (a, 0.). Answer: a