\documentclass[oneside]{amsart}

% Boilerplate to put your name on every page
% This makes grading easier
\usepackage{authoraftertitle}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyfoot{}
\lhead{\MyAuthor}
\rhead{\thepage}

% Use letters in enumerated lists
\usepackage{enumitem}
\setenumerate[0]{label=(\alph*)}

% You can define abbreviations here
\newcommand{\problem}[1]{\par\noindent\textbf{Problem~#1}}
\newcommand{\answer}[1]{{\par\medskip\sffamily#1\par\medskip}}
\newcommand{\cl}{\mathop{\mathrm{cl}}\nolimits}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}

\begin{document}

% Put your title, name, and the date here
\title{Homework 8}
\author{Your Name Here}
\date{April 6, 2020}

\maketitle


\problem{2.2.10.} Are there any values of $n$ in $\N$ such that $f(x)=|x|x^n$ is differentiable at $x=0$?

\answer{Type your answer here.}

\clearpage

\problem{2.3.3.} Let $f:[0,\infty)\to \R$ be a continuously differentiable function such that $f(0)>0$ and there is a constant $C$ with $|f'(x)|\leq C<1$ for all $x$ in $(0,\infty)$.

\begin{enumerate}
    \item Use the Mean Value Theorem to show that  $f(x)\leq f(0)+Cx$ for all $x\geq 0$.
    
    \item Prove that $\lim_{x\to\infty} [f(x)-x]=-\infty$.

    \item Show that there is at least one $x_0>0$ such that $f(x_0)=x_0$.

    \item Prove that there is at most one $x_0>0$ (thus exactly one $x_0>0$) such that $f(x_0)=x_0$.
\end{enumerate}

\answer{
\begin{enumerate}
    \item Answer to part (a)
    \item Answer to part (b)
    \item
    \item
\end{enumerate}
}

\clearpage

\problem{2.3.8.} Why can't you prove Proposition 2.3.10 by applying the Chain rule to the fact that $f(f^{-1}(\zeta))=\zeta$ for all $\zeta \in (\alpha,\beta)$?

\answer{Type answer here.}

\end{document}