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.