PROBLEM SET VI
“DISTINGUISHING”
DUE FRIDAY, 28 OCTOBER
Exercise 41. Consider the function f : R
R given by f (x) = x 1/3 . Note that f is continuous, and show that
f is differentiable at all points x ∈ R such that x 6= 0. Show that f is not differentiable at 0 according Df. 4.4 in
Apostol. Nevertheless, make precise
pthe idea that there is a tangent line to f at 0, and its slope is infinite. Does your
idea apply to the function g (x) := |x| as well?
R is said
Definition. Suppose U ⊂ R an open set, and suppose k a natural number. Recall that a function f : U
to be of class C k if for every natural number n ≤ k, the n-fold derivative f (n) exists and is continuous on U . We
denote by C k (U ) the set of such functions.
A function f : U
R is said to be smooth or infinitely differentiable or of class C ∞ if it is of class C k for every
natural number k. We denote by C ∞ (U ) the set of such functions.
Exercise 42. Suppose n a positive integer. Consider the function fn : R
R defined by
n
fn (x) := |x| .
What is the largest natural number k such that fn ∈ C k (R)?
Definition. We will now define, for any natural number n, a polynomial pn . We do this recursively. First, set
p0 (x) := 1
p1 (x) := x.
and
Now for any n ≥ 1, set
pn+1 (x) := 2x pn (x) − pn−1 (x).
R.
We will regard these polynomials as functions R
Exercise 43. Let’s establish some basic properties of the polynomials pn .
(a) Compute pn for 0 ≤ n ≤ 6 explicitly.
(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
p m ( pn (x)) = p mn (x).
Exercise 44. Show that for any integer n ≥ 0 and for any x ∈ R, one has
pn0 (x) = 2n
n/2
X
if n is even and
pn0 (x) =
p2i−1 (x)
i =1
n + 2n
bn/2c
X
i =1
if n is odd.
1
p2i (x)
2
DUE FRIDAY, 28 OCTOBER
Definition. Let us say that a function φ: R
R is of C-type if for any integer n ≥ 0 and any x ∈ R, one has
φ(n x) = pn (φ(x)).
Exercise 45. Suppose φ, ψ: R
R are functions such that for any x, y ∈ R, one has
φ(x + y) = φ(x)φ(y) − ψ(x)ψ(y),
ψ(x + y) = φ(x)ψ(y) + ψ(x)φ(y),
and
φ(x)2 + ψ(x)2 = 1.
Show that φ is of C-type.
Exercise 46. Suppose φ: R
R a function of C-type. Show that for any integer n ≥ 1 and any x ∈ R, one has
φ0 ((n + 1)x) − φ0 ((n − 1)x) = 2φ(n x)φ0 (x).