Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 9
Exercise 1 Prove that x7→ x|x|is differentiable on R.
Exercise 2 Let f:I→R, where Iis a real interval, be a differentiable function.
(1) We assume that the derivative of fis bounded on I. Prove that fis C-Lipschitz on I.
(2) Prove that if fis C-Lipschitz on Ithen its derivative is bounded on I.
(3) Give an example of a C-Lipschitz function which derivative is not bounded (of course consider a
non differentiable function).
Exercise 3 Let f: [0,1] →[0,1] be a continuous function. We assume that f(0) >0and f(1) <1.
Prove, using the intermediate value theorem that fadmits a fix point (ie, a∈[0,1] such that f(a) = a).
Exercise 4 Let P:R→Rbe a polynomial function of degre n,P(x) = a0+a1x+a2x2+· · · +anxn.
Compute its derivative of order kat the origin 0, for every positive integer k.
Exercise 5 Prove that sin x≤xfor nonnegative x.
1