\documentclass[12pt]{amsart}

\usepackage{amsmath, amssymb, amsthm}
\usepackage[margin=0.5in]{geometry}
\usepackage[pdflang={en-US}]{hyperref}
\usepackage[shortlabels]{enumitem}
\usepackage{multicol}

\usepackage{tikz}

\renewcommand{\thesection}{\Alph{section}}

\newcommand{\scs}[1]{\begin{scshape}#1\end{scshape}}
\newcommand*{\inv}{^{-1}}
\newcommand*{\RR}{\mathbb{R}}
\newcommand*{\QQ}{\mathbb{Q}}
\newcommand*{\ZZ}{\mathbb{Z}}
\newcommand*{\NN}{\mathbb{N}}

\newcommand*{\inte}[2][x]{\int #2\, d#1}
\newcommand*{\der}[2][x]{\frac{d}{d#1}\!\left[ #2 \right]}
\newcommand*{\dery}{\frac{\text{d}y}{\text{d}x}}

\newtheorem*{theorem}{Theorem}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}
\newtheorem{problem}{}
\newtheorem*{challenge}{Challenge}
\newtheorem*{example}{Example}
\newtheorem*{solution}{Solution}
\newtheorem*{ruleofthumb}{Rule of Thumb}
\newtheorem*{method}{Method}

\newcommand{\thm}[1]{\begin{theorem}#1\end{theorem}}
\newcommand{\prob}[1]{\begin{problem}#1\end{problem}}
\newcommand{\dfn}[1]{\begin{definition}#1\end{definition}}
\newcommand{\chal}[1]{\begin{challenge}#1\end{challenge}}
\newcommand{\eg}[1]{\begin{example}#1\end{example}}
\newcommand{\sol}[1]{\begin{solution}#1\end{solution}}
\newcommand{\rot}[1]{\begin{ruleofthumb}#1\end{ruleofthumb}}
\newcommand{\mtd}[1]{\begin{method}#1\end{method}}

\newcommand{\abs}[1]{\left| #1 \right|}

\title{The shapes of graphs and absolute extremes}
\date{October 31, 2022}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{

Sketch the graph of a function $y=f(x)$ such that all of the following are true:
\begin{enumerate}[a)]
\item $f'(-2) =0$ and $f''(-2) > 0$
\item $f(0)= 1$, $f'(0)= 0$, and $f''(0) < 0$
\item $f'(3) > 0$ and $f''(3) < 0$
\item $\lim_{x \to \infty} f(x) = 0$
\end{enumerate}

\begin{tikzpicture}
\draw[step=0.8cm,color=lightgray,very thin] (-6.5,-4.9) grid (6.5,4.9);
\end{tikzpicture}
}

\prob{
Why does $f(x) = x - \frac{1}{x}$ not have an absolute minimum on the interval $(0,2]$?
Does it have an absolute maximum?
}

\newpage

\prob{
Find the absolute minimum and maximum of $f(x) = x-3x^{2/3}$ over the interval $[ -1,10]$.
It may help to know that $f(10) \approx -3.9248$.
}

\end{document}
