Just let b=4 is ok

(a) Assume the third order derivative of y = f(x) exists, and a + f(x) + sin(f(x)) = 0. Evaluate f'" (0). [4 marks (b) Evaluate the following limits: [8 marks] (i) lim arccos(x - 1) In(x) x-1- arccos (x) (ii) lim (tan x + sinx) (72) (tan (2.) ) 999 (iii) lim 499 (sin (4x)) 500 (iv) lim ((x + 1) 10000 In(5x + 5)) x--1+ (c) Let f(x) = sinh(x) + tanh(x), where sinh(x) = , and tanh(x) = eater. el-e - Given that f(x) is one-to-one, use the linear approximation of f- around a = 0 to estimate f-1(0.01 x (b + 1)), where b denotes the last digital number of your student id number (for example, if your student number is 20987654D, then b = 4). [3 marks] (d) If f(xo) = To, then To is a fixed point of f. Suppose that f is differentiable in [0, 1], and both 0 and 1 are fixed points of f. Besides, f'(0) > 1 and f'(1) > 1. Prove that (i) f has at least one fixed point in (0, 1). [2 marks] (ii) There exists a fixed point & E (0, 1) such that f'(8)