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\Huge{Extrema \& Inflection Point}\\
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\normalsize
Note: a personal copy of this worksheet may be used during the quiz next Tuesday.


\begin{enumerate}
\item Let $f(x)=x\ln(x^2)$.  

\begin{enumerate}
\item Find the critical points of $f$.
\vfill
\item Find the $x$ and $y$ values of all the local maximums of $f$.
\vfill
\item Find all inflection points of $f$.
\vfill
\end{enumerate}


\item Consider the parabola determined by $y=ax^2+bx+c$ where $a$, $b$, and $c$ are real numbers.

\begin{enumerate}

\item Use calculus to find the critical points of the parabola.  (Remember, $a$, $b$, and $c$ are to be treated as constants. 
\vfill
\item Does your answer from above seem familiar?  What is another name for the point that you just found?
\end{enumerate}


\newpage 
\item Sketch a graph of a function $\alpha$ that satisfies \emph{all} of the following:
\begin{itemize}
\item $\alpha'(x)<0,$ when $x<-3$ 
\item $\alpha'(x)\geq0,$ when $-3<x<2$
\item $\alpha'(x)=-2$ when $2<x$
\item $\alpha'(2)$ is not defined
\item $\alpha'(-3)=\alpha'(1)=0$
\item $\alpha''(x)>0$, when $x<-2$ \\ \& when $1<x<2$
\item $\alpha''(x)<0$, when $-2<x<1$
\end{itemize}




\vspace{-2in} \hspace{2.5in} \includegraphics[height=2.3in]{blank_5}	



\item Consider $g(x)=x^4$.  Use $g$ to show that the roots of $g''$ are not necessarily points of inflection.
\vskip 2in
\item Let $\beta$ be the function whose \emph{derivative} is shown below.  Find:

\begin{enumerate}
\item the critical points of $\beta$.
\vfill
\item the $x$ coordinates of \\ any local maximums.
\vfill
\item the $x$ coordinates of \\any points of inflection
\vfill
\end{enumerate}

\vspace{-2.5in} \hspace{2.1in}  \includegraphics[height=2in]{4poly}









\end{enumerate}
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