3. Let X1, X2, ..., Xbe iid having the common pdf S 2/r if l<r< , f(1) = 0 elswhere. Is there a real number a such that X a as n o ?
Let S function, f: S R, between the two sets. x < 1}. Show that S and R have the same cardinality by constructing a bijective x E R 0
Let f (2) be defined by: k-?, <<-1 f(3) = z? +, -1<x<1 - kr1 Which of the following values of k would make f (2) continuous on R? Ok=0 There is no such value for k Ok= -1 Ok= 1
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
Convex Optimization
Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
1. Let x, a € R. Prove that if a <a, then -a < x <a.
(a) Let x(t) = 1 when 0 <t<1 and 0 for all other real t. Find and graph the following: (i) r(t -3). [5] (ii) c(t/2). (5] (iii) <((t-3)/2). [5] (iv) (t/2) – 3). [5]
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.