MATH 295 PROBLEMS 9

IGOR KRIZ

1. Construct a continuous onto map f : [0, 1) → (−1, 1). [Hint: f is not injective.]

2. Prove that

( \begin{matrix}n + 1 \\ r + 1 \end{matrix} ) = ( \begin{matrix}n \\ r \end{matrix} ) + ( \begin{matrix}n − 1 \\ r \end{matrix} ) + ... + ( \begin{matrix}r \\ r \end{matrix} ) .

3. The function f(x) = \frac{(1+x)^{n}−1}{x} is defined for all x ≠ 0. Define f(0) in such a way that f becomes a continuous function defined on ℝ.

4. Let f(x) be any polynomial function [this means f(x) = a_{n}x^{n} + a_{n−1}x^{n−1} + ... + a_{0}]. Prove that there is some number y such that |f(y)| ≤ |f(x)| for all x.

5. Prove that there exists a number x such that

\frac{1}{x} = \sqrt{x} + \sqrt[3]{x}.

[Hint: don’t try to solve the equation explicitly.]