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\title{Extremes}
\date{October 19, 2021}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\dfn{
The number $a$ (in the interior of the domain of $f$) is a \textbf{critical number} for $f(x)$ if $f'(a) = 0$  or $f'(a)$ does not exist.
}

\prob{
Sketch the graph of a function that has a local maximum at $x=-5$, a critical number that is not a local extreme at $x=-1$, and local minimum at $x=3$.

\begin{center}
\begin{tikzpicture}
\draw[step=0.5cm,color=lightgray,very thin] (-6,-3) grid (6,3);
\draw[<->,very thick] (-6,0)--(6,0);
\draw (6,0) node[anchor=north east]{$x$};
\draw[<->,very thick] (0,-3)--(0,3);
\draw (0,3) node[anchor=north east]{$y$};
\end{tikzpicture}
\end{center}
}

\prob{
Find the critical numbers of the function: \[ g(x) = 3x^4 - 8x^3 + 6x^2 - 2\]

\vfill
}



\prob{
Find the critical numbers of the function (it may help to sketch a graph of the function): \[f(x) = |x^2-2x|\].

\vfill
}

\newpage

\thm{[Extreme Value Theorem]
If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a,b]$.
}

\begin{method}
To find the absolute extreme values of a continuous function $f$ over the interval $[a,b]$:
\begin{enumerate}[1.]
\item Find the critical numbers for $f$ \textbf{that lie in the interval $[a,b]$}.
\item Evaluate the function at these critical numbers and at $a$ and $b$.
\item The largest value obtained in the previous step is the absolute maximum, the smallest is the absolute minimum.
\end{enumerate}
\end{method}

\prob{
Find the absolute maximum and minimum values of the function $g(x) = 3x^4 - 8x^3 + 6x^2 - 2$ over the interval $[-1, 2]$.

\vfill
}

\prob{
Find the absolute maximum and minimum values of the function $f(x) = |x^2-2x|$ over the interval $[0.5, 4]$.

\vfill
}

%\prob{
%Find the absolute maximum and minimum values of the curve  $y^2 + xy + x^2 = 3$ over the interval $[-1,1]$.
%}

\chal{
Prove that $f(x) = x^3 + x^2 + x + 1$ has no local extremes.
}

\end{document}