<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{amsart}

\usepackage{graphicx,subfig}
\usepackage{amscd}
\usepackage{xy}
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\usepackage{pinlabel}
\usepackage[linktocpage]{hyperref}
\usepackage[margin=1in]{geometry}
\usepackage{parskip}

\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
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\newcommand{\cpn}{\C\textup{P}^n}
\newcommand{\rpn}{\R\textup{P}^n}

\DeclareMathOperator{\image}{image}


\title{MAT 239: Homework 5}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item Let $f:\R^k\to \R$ be a smooth function. For each $a\in \R^k$ define
		$$f_a(x_1,\dots, x_k) = f(x)+a_1x_1+\cdots +a_kx_k.$$
		Prove that for almost every $a\in \R^k$ (in the sense of Sard's theorem), that $f_a$ is a Morse function. 
		
		Hint: show that you can apply the generic transversality theorem to the family of functions $F: \R^k\times \R^k \to \R^k$ defined by
		$$F(x,a) = \left( \frac{\partial f}{\partial x_1}(x)+a_1, \dots, \frac{\partial f}{\partial x_k}+a_k\right)$$
		
		\item Let $E=\{x^2+2y^2+3z^2=1\}$ and let $S_a = \{x^2+y^2+z^2=a\}$. Determine for which values of $a$, $E$ intersects $S$ transversally.
		
		\item Each of the following functions has a critical point a the origin. Determine whether or not this critical point is non-degenerate, and if it is, determine whether it is a local maximum (index 2), saddle (index 1), or local minimum (index 0).
		\begin{enumerate}
			\item $f(x,y) = x^2+4y^3$
			\item $f(x,y) = x^2-2xy+y^2$
			\item $f(x,y) = x^2+y^4$
			\item $f(x,y) = x^2+11xy+y^2/2+x^6$
			\item $f(x,y) = 10xy+y^2+75y^3$
		\end{enumerate}
		
	\end{enumerate}
\end{document}</pre></body></html>