Homework 14: The Intermediate Value Theorem

Instructions. Assignments should be stapled and written clearly and legibly. Problems 4 and 5 are optional.

1. Section 5.3, #5, 7, 10.
2. Let \(f\) be continuous on \([0,1]\) with \(f(0) = f(1)\). Prove that there exists \(c\in [0,\frac{1}{2}]\) such that \(f(c) = f(c + \frac{1}{2})\).
3. Prove that there exists a real number \(x\) such that \begin{equation} x^{177}+\frac{165}{1+x^{8}+\sin^2x} = 125. \end{equation}
4. Prove that if \(f:[a,b]\to \mathbb{R}\) is injective and continuous, then the inverse function \(f^{-1}\) is also continuous.
5. (Putnam Competition) Suppose that the real numbers \(a_0, a_1, \dots, a_n\) and \(x\), with \(0 \lt x \lt 1\), satisfy \[ \frac{a_0}{1-x} + \frac{a_1}{1-x^2} + \cdots + \frac{a_n}{1 - x^{n+1}} = 0. \] Prove that there exists a real number \(y\) with \(0 \lt y \lt 1\) such that \[ a_0 + a_1 y + \cdots + a_n y^n = 0. \]