All parts of this question concern the function f($) : 6 sin :B + 2 cos :12. (a) Find the smallest positive constant M that satisfies M Z ifik) (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 (b, one has f(:1:) = V 40 sin (a: -l- (I?) for all real :3. Answer: M : gamma) n\" for some 15 between 0 and at. This is (n + 1)! Recall the standard decomposition an) = Tubs) + En(x), in which Lagrange's formula says En(w) : valid for every integer n 2 0. In both parts below, estimate En(:c) using Lagrange's formula with the constant Mfound in part (3). (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 IEn(0.3)| S 0,5 x 1017. (C) Suppose n : 9 is prescribed. Find the largest positive number a. such that the approximation T9(:13) for f(a:) is guaranteed to be accurate to within 9 decimal places, for all :3 in the symmetric interval (*a, (1). Answer: a