Problem solving seminar VIII
28. For 0 ≤ x ≤ π/2 and 0 < p < 1 prove that
(cos x)p
≤ cos(px).
29. Define a sequence of positive numbers as follows: let x0 > 0 be any posive number
and xn+1 = (1 + xn)−1
. Prove that this sequence converges and find its limit.
30. Prove:
(1) If B is a real symmetric positive definite matrix, there is an invertible matrix
C such that B = CT
C.
(2) If A is a real symmetric n × n matrix find a maximum of the function
F(x) =
Ax, x
x, x
defined for x ∈ Rn
− {0} . In which x is this maximum attained?
(3) If A and B are real symmetric n × n matrices, B positive definite, then the
function
G(x) =
Ax, x
Bx, x
defined for x ∈ Rn
−{0} takes its maximum only in an eigenvector for a certain
matrix related to A and B. Show which matrix.
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