Analysis Homework #5 due Thursday, Feb. 3 1. Suppose that f is diﬀerentiable with f ′ (x) = x2 f (x), f (0) = 1. Show that f (x)f (−x) = 1 for all x ∈ R and that f (x) > 0 for all x ∈ R. 2. Suppose that f is diﬀerentiable with f ′ (x) = 2f (x) for all x. Show that there exists some constant C such that f (x) = Ce2x for all x. 3. Show that ex ≥ x + 1 for all x ∈ R. 4. Suppose that f is diﬀerentiable with |f (x) − f (y)| ≤ |x − y|2 for all x, y ∈ R. Show that f is actually constant. Hint: you need to show that f ′ (y) = 0 for all y. • 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.