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\begin{center}\textbf{Problem Set 7 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 10/28/24} \end{center}

\noindent Ten of these problems will be graded, with each problem worth 2 points. Clear and complete justification is required for full credit. You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor.

\begin{enumerate}
	\item  Let $f(x) = |x|$.  Find $f^\prime(0)$, or show it does not exist.
	\item  Let $f(x) = |x| \cdot x$.  Find $f^\prime(0)$, or show it does not exist.
	\item Let $f(x) = (x-2)^{2/3}$.  Find $f^\prime(2)$, or show it does not exist.
	\item Determine where the function \[f(x)=\begin{cases} x \sin \frac{1}{x}&\mathrm{if } \, x \neq 0\\
	\enspace 0&\mathrm{if } \, x=0 \end{cases}\]
	is differentiable.

	\item Determine where the function \[f(x)=\begin{cases} x^2 \sin \frac{1}{x}&\mathrm{if } \, x \neq 0\\
	\enspace 0&\mathrm{if } \, x=0 \end{cases}\]
	is differentiable.
	\item Prove the Sum Rule for Derivatives.  Make it clear why each hypothesis matters.
	\item Prove the Quotient Rule for Derivatives.  Make it clear why each hypothesis matters.
	\item Do Exercise 5.1.6.
	\item Do Exercise 5.1.9(a).
	\item Prove Corollary 5.2.2(a).
	\item Give an example of a function $f$ that is differentiable at $x = a$ such that $f'(a)$ exists, with  $f'(a) \neq 0$, but yet $f$ attains a relative extremum at $x = a$.
	\item Give an example of a function $f$ that is continuous at $x = a$, not differentiable at $x = a$, but yet $f$ attains a relative extremum at $x = a$.

\end{enumerate}

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