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\begin{document}

\begin{center}Math 335 Real Analysis

Exam II

October 17, 2001

\end{center}

\textbf{Answer all 5 questions.  All questions have equal points. 
Symbols and markings without complete sentences will not be considered as
answers.}

\begin{enumerate}
\item State the Intermediate Value Theorem (I.V.T.).
\begin{itemize}

\item[{a)}] State clearly a consequence of the IVT for any odd degree
polynomial
$P(x)$.

\item[{b)}] Specify a closed interval $[a, b]$ on which the polynomial

$P(x) = 8x^3 - 36x^2 + 46x - 15 $

must have a root.  Give your reasons.
\end{itemize}


\vbox to 5.5in{}
\pagebreak

\item Let $S$ be a subset of $\mathbb{R}$.
\begin{itemize}
\item[{a)}]  Define what is meant by saying $S$ is bounded above.

\item[{b)}]  Define what is meant by saying $U$ is a least upper bound of
$S$.

\item[{c)}]  State a theorem which guarantees the existence of a least
upper bound.

\item[{d)}]  Give the least upper bound and the greatest lower bound of
the set formed by the terms of the sequence $\{{S_n}\} = \{{(-1)^n + 1 +
\frac{1}{n}}\}^\infty_{n = 1}
$
\end{itemize}


 
\pagebreak
\item The double inequality
$-1 \leq \sin \frac{1}{x} \leq 1 $
holds for all $x \neq 0. $  Use the Sandwich Theorem to show

$\lim\limits_{x \to 0} x \sin \frac{1}{x} = 0.$

\

Determine whether the function

$$f (x) = \left\{ \begin{array}{l}
x^2 \sin \frac{1}{x} \hskip1in x \neq 0 \\
\ \ \ 0 \hskip1.28in x = 0
\end{array} 
\right.$$

has a derivative at $x = 0 $ and if it does have a derivative, determine
its value.

\pagebreak
\item Let $f$ be a function defined on an open interval I which has an
absolute maximum value occurring at $x_0  \epsilon I.$  If $f' (x_0)$
exists, prove that $f' (x_0) = 0. $

\pagebreak
\item State the Mean Value Theorem (M.V.T.).  By carefully applying the
M.V.T., show that

$\sin x > x \cos x $ when $0 < x < \pi. $

Hint:  Take $f (x) = \sin x - x \cos x. $
\vskip.15in


\end{enumerate}



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