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%  Extreme Values: The First Derivative Test            %
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\problemgrouptitle{Extreme Values: The First Derivative Test}

\begin{problemblock}[2]{Find the open intervals upon which the following functions are increasing or decreasing.  Also find any \relative minima and maxima and their locations.}

\item $f(x)=x^2+2x+1$
\answer{Decreasing on: $(-\infty,-1)$; Increasing on: $(-1,\infty)$; No \relative maxima; \Relative minimum: $f(-1)=0$}

\item $f(x)=\displaystyle\frac{x^2-3x+1}{x-1}$
\answer{Increasing on: $(-\infty,1)\cup(1,\infty)$; No \relative maxima or minima}

\item $f(x)=4\cos x-2x$ on $[0,2\pi]$
\answer{Decreasing on: $\left(0,\displaystyle\frac{7\pi}{6}\right)\cup\left(\displaystyle\frac{11\pi}{6},2\pi\right)$; Increasing on: $\left(\displaystyle\frac{7\pi}{6},\frac{11\pi}{6}\right)$;\newline \Relative maximum: $\displaystyle f\left(\frac{11\pi}{6}\right)=2\sqrt{3}-\frac{11\pi}{3}$; \Relative minimum: $\displaystyle f\left(\frac{7\pi}{6}\right)=-2\sqrt{3}-\frac{7\pi}{3}$}

\item $f(x)=|x|$
\answer{Decreasing on: $(-\infty,0)$; Increasing on: $(0,\infty)$; \Relative minimum: $f(0)=0$}
\item $\displaystyle f(x)=\left\{\begin{array}{cl}
|x| & \mbox{if}\ x\neq 0 \\[1em]
2 & \mbox{if}\ x=0
\end{array}\right.$
\answer{Decreasing on: $(-\infty,0)$; Increasing on: $(0,\infty)$; \Relative maximum: $f(0)=2$.  Note that the First Derivative Test cannot be applied here because $f(x)$ is discontinuous at $x=0$.  One must return to the definition of \relative maximum to evaluate the critical number $x=0$\,.}

\end{problemblock}

\textbook{\addtoanswerkey{\clearpage}}
\lab{\addtoanswerkey{\clearpage}}
\begin{problemblock}[2]{Each graph below is a graph of the derivative $f'$ of a function $f$.  In each case use the graph of $f'$ to sketch a possible graph of $f$.}

\addtoanswerkey{\begin{multicols}{2}}
\item ~

\resizebox{0.80\columnwidth}{!}{%
\includegraphics{graphics/first-derivative-test1}%
}
\answer{~

\resizebox{0.55\columnwidth}{!}{%
\includegraphics{graphics/first-derivative-test2}%
}
}

\item ~

\resizebox{0.80\columnwidth}{!}{%
\includegraphics{graphics/first-derivative-test3}%
}
\answer{~

\resizebox{0.55\columnwidth}{!}{%
\includegraphics{graphics/first-derivative-test4}%
}
}

\end{problemblock}

\addtoanswerkey{\end{multicols}Note that your graphs should be equivalent up to vertical shift by a constant.}

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