Problem solving seminar II
6. Let the real valued function f be defined in an open interval about the point a on
the real line and be differentiable at a. Prove that if (xn) is an increasing sequence
and (yn) is a decreasing sequence in the domain of f, and both sequences converge to
a, then
lim
n→∞
f(yn) − f(xn)
yn − xn
= f (a).
7. Let α1, α2, . . . , αn be distinct real numbers. Show that the n exponential functions
eα1t
, eα2t
, . . . , eαnt
are linearly independent over the real numbers.
8. Let A and B be n × n complex unitary matrices. Prove that
| det(A + B)| ≤ 2n
.
9. Suppose that f is continuous real valued function. Show that
1
0
f(x)x2
dx =
1
3
f(ξ)
for some ξ ∈ [0, 1].
Homework II. Does every positive polynomial in two variables take its minimum in
the plane?
1