<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\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	$f(x) = \sin(x), \quad 0 \leq x &lt; \pi/2$.
			\item	$f(x) = \ln(x), \quad 0&lt;x \leq 2$.
			\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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