Analysis & Group theory problems
1. Find the minimum value of f (x) = (2x2 − 5x + 2)3 over the closed interval [0, 1].
2. Show that log x ≤ x − 1 for all x > 0.
3. Suppose that x > y > 0. Use the mean value theorem to show that
1−
x
y
< log x − log y < − 1.
x
y
4. Let ∗ be an operation on a set S. Suppose that ∗ is associative, and that S contains
an identity element for ∗. Prove that if x, y are two elements of S which are invertible
with respect to ∗, then x ∗ y is invertible with respect to ∗. What is (x ∗ y)−1 ?
5. Let GL(2, R) denote the set of invertible 2 × 2 matrices with real entries. Which of the
following is a group operation on GL(2, R)?
(a). matrix addition
(b). matrix multiplication
(c). the operation ∗ given by A ∗ B = AB − BA for matrices A and B