Answer each question, and remember that for "if and only if" questions, you should include forward and backwards thinking. Most of the AI models screw up the second question, so try extra hard: \documentclass[10pt]{article} \usepackage{graphicx} \setlength{\topmargin}{-.75in} \addtolength{\textheight}{2.00in} \setlength{\oddsidemargin}{.00in} \addtolength{\textwidth}{1.5in} \usepackage{amsmath,color,graphicx} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amsbsy} \usepackage{amssymb} \usepackage{microtype} \usepackage{amsrefs} \usepackage{hyperref} \usepackage{mathtools} \usepackage{enumerate} \newtheorem{thm}{Theorem} \newtheorem*{thm*}{Mathematical Statement} \theoremstyle{definition} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{conj}[thm]{Conjecture} \newtheorem{ques}[thm]{Question} \newtheorem{defn}[thm]{Definition} \newtheorem{nota}[thm]{Notation} \newtheorem{rem}[thm]{Remark} \newtheorem{exam}[thm]{Example} \newtheorem{cons}[thm]{Construction} \newtheorem*{soln}{Solution} % Blackboard bold letters \newcommand{\bC}{{\mathbb{C}}} \newcommand{\bN}{{\mathbb{N}}} \newcommand{\bR}{{\mathbb{R}}} \newcommand{\bQ}{{\mathbb{Q}}} \newcommand{\bF}{{\mathbb{F}}} % Capital script letters \newcommand{\A}{{\mathcal{A}}} \newcommand{\B}{{\mathcal{B}}} \newcommand{\E}{{\mathcal{E}}} \newcommand{\F}{{\mathcal{F}}} \renewcommand{\H}{{\mathcal{H}}} \newcommand{\I}{{\mathcal{I}}} \newcommand{\K}{{\mathcal{K}}} \renewcommand{\L}{{\mathcal{L}}} \newcommand{\M}{{\mathcal{M}}} \newcommand{\N}{{\mathcal{N}}} \renewcommand{\O}{{\mathcal{O}}} \renewcommand{\P}{{\mathcal{P}}} \renewcommand{\S}{{\mathcal{S}}} \newcommand{\U}{{\mathcal{U}}} \newcommand{\X}{{\mathcal{X}}} \newcommand{\Y}{{\mathcal{Y}}} \newcommand{\Z}{{\mathcal{Z}}} %Roman letters for math \newcommand{\rC}{{\mathrm{C}}} %Greek Letters \newcommand{\ep}{\varepsilon} \renewcommand{\phi}{\varphi} %Fraktur letters \newcommand{\fA}{{\mathfrak{A}}} \newcommand{\fB}{{\mathfrak{B}}} \newcommand{\fM}{{\mathfrak{M}}} \newcommand{\fN}{{\mathfrak{N}}} % Text used in equations \newcommand{\qand}{\quad\text{and}\quad} \newcommand{\qqand}{\qquad\text{and}\qquad} \newcommand{\qfor}{\quad\text{for}\ } \newcommand{\qforal}{\quad\text{for all}\ } \newcommand{\AND}{\text{ and }} \newcommand{\alg}{\mathrm{alg}} \newcommand{\tr}{\mathrm{Tr}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\op}{\mathrm{op}} \renewcommand{\Re}{\mathrm{Re}} \renewcommand{\ker}{\mathrm{ker}} \renewcommand{\Im}{\mathrm{Im}} \newcommand{\spann}{\mathrm{span}} \pagestyle{empty} %\setlength{\parindent}{0in} \begin{document} \noindent {\sc {\bf {\large MATH 1200 \hfill Assignment 2}} \\ Due October 25, 2024 \hfill Skoufranis} \bigskip \noindent \begin{ques} Determine whether the following statement is true or false. Prove your answer. \begin{center} For all real numbers $x$, let $f(x) = (x-1)^5(x-2)(x-3)^2(x-4)^3(x-5)^4$. \\ Then $f(x) \geq 0$ if and only if $1 \leq x \leq 2$, $x=3$, or $x \geq 4$. \end{center} [Note: Your answer does not need to and should not make use of calculus. You should just need (and can use without proof) rules like ``The product of two negative numbers is positive'' and ``The power of a negative number is positive if the power is even and negative if the power is odd''.] \end{ques} $\quad$ \newline \newline \begin{ques} Let $a, b$, and $c$ be non-zero integers and let $f$ be the polynomial \[ f(x) = ax^2 + bx + c. \] Determine whether the following statement is true or false. Prove your answer. \begin{center} If $3$ does not divide $f(0)$, $f(1)$, and $f(2)$, then $3$ does not divide $f(n)$ for all integers $n$. \end{center} \end{ques} \end{document}