Analysis Homework #6 due Thursday, Feb. 17 1. Suppose ∫ 1 f is non-negative and integrable on [0, 1] with f (x) = 0 for all x ∈ Q. Show that 0 f (x) dx = 0. 2. Suppose f is increasing on [0, 1]. Show that f is integrable on [0, 1]. Hint: let P be the partition of [0, 1] into n subintervals of the same length and show that S + (f, P ) − S − (f, P ) = f (1) − f (0) . n 3. Show that there exists a unique function f , defined for all x ∈ R, such that f ′ (x) = 1 , 1 + x4 f (0) = 0. 4. Suppose f is continuous on [a, b]. Show that there exists some c ∈ (a, b) such that ∫ b f (t) dt = (b − a) · f (c). a Hint: apply the mean value theorem to the function F (x) = ∫x a f (t) dt. • You are going to work on these problems during your Thursday tutorial (at 4 pm). • When writing up solutions, write legibly and coherently. Use words, not just symbols. • Write your name and then MATHS/TP/TSM on the first page of your homework. • Your solutions may use any of the results stated in class (but nothing else). • NO LATE HOMEWORK WILL BE ACCEPTED.