PROBLEM SET VI
“WHERE DO YOU THINK YOU’RE GOING?”
DUE FRIDAY,  OCTOBER 
Exercise . If a differentiable function g : R . R is bounded, must its derivative
g′ : R . R be bounded? Prove or give a counterexample.
Exercise . Suppose g : [0, ∞) . R a continuous function that is differentiable
on (0, ∞). Show that if g(0) = 0 and if, for any t > 0, one has |g′ (t)| ≤ |g(t)|,
then in fact g = 0.
Exercise⋆ . Suppose g : R . R a differentiable function whose derivative is
bounded by a fixed constant C > 0. (at is, assume that for any x ∈ R, one has
|g′ (x)| ≤ C.)
(a) If C < 1, show that there exists a real number x such that g(x) = x.
(b) Find an example for C = 1 such that there is no x ∈ R for which g(x) = x.
(You may use use any function you like to construct this counterexample,
even if we haven’t discussed it in class.)
Definition. Suppose U ⊂ R an open set, and suppose k a natural number. Recall
that a function g : U . R is said to be of class Ck if for every natural number
n ≤ k, the n-fold derivative g(n) exists and is continuous on U. We denote by
Ck (U) the set of all such functions.
A function f : U . R is said to be smooth or infinitely differentiable or of class
C∞ if it is of class Ck for every natural number k. We denote by C∞ (U) the set of
such functions.
Exercise . Suppose n a positive integer. Consider the function gn : R .
fined by
gn (x) := |x|n .
What is the largest natural number k such that gn ∈ Ck (R)?
R de-
Definition. We will now define, for any natural number n, a polynomial pn . We
do this recursively. First, set
p0 (x) := 1 and p1 (x) := x.
Now for any n ≥ 2, set
pn (x) := 2xpn−1 (x) − pn−2 (x).
We will regard these polynomials as functions R .

R.

DUE FRIDAY,  OCTOBER 
Exercise . Let’s establish some basic properties of the polynomials pn .
(a) Compute pn for 0 ≤ n ≤ 6 explicitly. (You may use a computer for this
computation if you wish.)
(b) Formulate a conjectural formula for the coefficient of the highest power of
pn . Prove your conjecture.
(c) Show that the constant term pn (0) is given by the formula
{
(−1)k if n = 2k;
pn (0) =
0
if n is odd.
(d) Show that for any natural number n, one has
pn (1) = 1
pn (−1) = (−1)n .
(e) Show that for any integers m, n ≥ 0 and any x ∈ R, one has
pm (pn (x)) = pmn (x).
Exercise . Show that for any integer n ≥ 0 and for any x ∈ R, one has
p′n (x)
= 2n
n/2
∑
p2i−1 (x)
i=1
if n is even and
⌊n/2⌋
p′n (x)
= n + 2n
∑
p2i (x)
i=1
if n is odd.
Definition. Let us say that a function φ : R .
n ≥ 0 and any x ∈ R, one has
R is of C-type if for any integer
φ(nx) = pn (φ(x)).
Exercise . Suppose φ, ψ : R .
has all of the following:
R are functions such that for any x, y ∈ R, one
φ(x + y) = φ(x)φ(y) − ψ(x)ψ(y)
ψ(x + y) = φ(x)ψ(y) + ψ(x)φ(y)
2
φ(x) + ψ(x)2 = 1.
Show that φ is of C-type.
Exercise . Suppose φ : R . R a function of C-type. Show that for any integer
n ≥ 1 and any x ∈ R, one has
φ′ ((n + 1)x) − φ′ ((n − 1)x) = 2φ(nx)φ′ (x).