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\sethandouttitle{Maximum and Minimum Values}
\allhandoutinfo{}
\begin{enumerate}
\item Sketch the graph of a function $f$ that is continous on $[1,5]$ and has an absolute minimum at 2,
absolute maximum at 3, and a local minimum at 4.
\item \begin{enumerate}
\item Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2.
\item Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2.
\item Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.
\end{enumerate}
\item Sketch the graph of $f$ and use your sketch to find the absolute and local maximum and minimum values of $f$.
\begin{enumerate}
\item $\displaystyle f(x) = \frac{1}{2}(3x-1), \quad x \leq 3$.
\item $f(x) = \sin(x), \quad 0 \leq x < \pi/2$.
\item $f(x) = \ln(x), \quad 0<x \leq 2$.
\item $f(x) = 1 - \sqrt{x}$.
\end{enumerate}
\item Find the critical numbers of the function.
\begin{enumerate}
\item $\displaystyle f(x) = 4+ \frac{1}{3}x - \frac{1}{2}x^2$.
\item $\displaystyle g(y) = \frac{y-1}{ y^2 - y + 1 }$.
\item $f(\theta) = 2 \cos \theta + \sin^2 \theta$.
\item $f(\theta) = x^2 e^{-3x}$.
\end{enumerate}
\item Find the absolute maximum and minimum values of $f$ on the given interval.
\begin{enumerate}
\item $f(x) = 12+4x-x^2$ on $[0,5]$.
\item $f(t) = t \sqrt{4-t^2}$ on $[-1,2]$.
\item $f(t) = 2 \cos t + \sin 2t$ on $[0,\pi/2]$.
\item $f(x) = x e^{-x^2/8}$ on $[-1,4]$.
\item $f(x) = \ln(x^2+x+1)$ on $[-1,1]$.
\end{enumerate}
\end{enumerate}
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