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%  Local and Global Extreme Values           %
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\problemgrouptitle{\Relative and \Absolute Extreme Values}

\item Identify the \relative maxima, \relative minima, \absolute maxima, and \absolute minima on each of the following graphs.
\begin{multicols}{3}
\begin{enumerate}
\item ~

\resizebox{0.95\columnwidth}{!}{
\asycode{%
import graph;
size(200,0);
real f(real x) {return 0.4*((x+2)^2*(x+.5)^2*(x-1)*(x-1.5)*(x-2)+5);}
pair F(real x) {return (x,f(x));}
draw(graph(f,-2.25,2.05,operator .., n=200),red+2);
xaxis(shift(-7,-10)*Label("$x$",align=E),xmax=2.5,Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=1,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(0)),above=true);
yaxis(shift(-8,-7)*Label("$y$",align=N),ymax=3.5,Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=1,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(0)),above=true);
dot(F(-.5),4+black);
dot(F(-.5),3+white);
dot(F(0.4),4+black);
dot(F(0.4),3+white);
dot(F(0.4)+(0,1),4+black);
dot(F(0.4)+(0,1),3+red);
}
} 

\item ~

\resizebox{0.95\columnwidth}{!}{
\asycode{%
import graph;
size(200,0);
real f(real x) {return -abs(x)+0.5;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-1.2,1.2,operator ..),red+2);
xaxis(shift(-7,-10)*Label("$x$",align=E),Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=0.5,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(0)),above=true);
yaxis(shift(-8,-7)*Label("$y$",align=N),ymin=-1.2,ymax=1.2,Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=0.5,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(0)),above=true);
}
}

\columnbreak
\item ~

\resizebox{0.95\columnwidth}{!}{
\asycode{%
import graph;
size(200,0);
real f(real x) {return -2/(x+8/3);}
pair F(real x) {return (x,f(x));}
real g(real x) {return (x+1)^2+2;}
pair G(real x) {return (x,g(x));}
real h(real x) {return -2/x;}
pair H(real x) {return (x,h(x));}
draw(graph(f,-8,-3,operator ..),red+2);
draw(graph(g,-3,0,operator ..),red+2);
draw(graph(h,0.25,4,operator ..),red+2);
xaxis(shift(-7,-10)*Label("$x$",align=E),xmax=6,Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=2,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(-8,6)),above=true);
yaxis(shift(-8,-7)*Label("$y$",align=N),ymax=8,Arrow(HookHead,size=4),ticks=Ticks(n=1,Step=2,Size=3,Label(fontsize(10)),OmitFormat(0),OmitTick(-8,8)),above=true);
dot(G(0),4+black);
dot(G(0),3+red);
dot(H(4),4+black);
dot(H(4),3+red);
}
}
\end{enumerate}
\end{multicols}
\answer{Note: It is assumed that the functions extend beyond the graph of the plot with the same trend unless they are explicitly terminated with a point. The values below are approximate.  
\begin{enumerate}
\item \Relative minima: $f(-1.4)=-0.8$ and $f(1.9)=0.5$; \Relative maxima: $f(-2.0)=2.0$, $f(0.4)=1.0$, and $f(1.3)=2.6$.  No \absolute minimum nor \absolute maximum.
\item \Absolute maximum value of $f(0)=0.5$~.  This is also a \relative maximum.  
\item \Relative minimum: $f(-1)=2$; \Relative maximum: $f(-3)=6$, $f(0)=3$. \Absolute maximum: $f(-3)=6$, No \absolute minimum.
\end{enumerate} 
}

\begin{problemblock}[2]{Find the critical numbers of the given function.}

\item $f(x)=x^3-9x^2+24x-15$
\answer{$x=2,4$}

\item $h(x)=|x|+1$
\answer{$x=0$}

\item $f(s)=\displaystyle\frac{s}{s^2+6}$ on the interval $[0,10]$.
\answer{$s=\sqrt{6}$}

\item $\displaystyle f(x)=x^3+5x^2+3x+1$
\answer{$\displaystyle x=-\frac{1}{3},\ -3$}

\item $\displaystyle g(t)=\frac{1}{4}t^4+2t^2-5t+6$
\answer{$\displaystyle t=1$}

\item $\displaystyle H(x)=\frac{x+3}{x-5}$
\answer{No critical numbers}

\item $\displaystyle f(t)=\sqrt{t^2-4}$
\answer{$\displaystyle t=-2,\ 2$}

\item $\displaystyle g(x)=\sqrt[3]{x^2-5}$
\answer{$\displaystyle x=-\sqrt{5},\ 0,\ \sqrt{5}$}

\item $\displaystyle F(\theta)=2\sin(\theta)-\theta$
\answer{$\displaystyle \theta\ \mbox{in}\ \left\{\frac{\pi}{3}+2n\pi\ |\ n\ \mbox{an integer}\right\}\bigcup\left\{\frac{5\pi}{3}+2n\pi\ |\ n\ \mbox{an integer}\right\}$}

\end{problemblock}

\begin{problemblock}{Find the \absolute maximum and \absolute minimum
values and their locations for the given function on the closed interval.}

\item $f(x)=x^4-x^2+1$ on $[-2,2]$
\answer{\Absolute maximum: $\displaystyle f(-2)=f(2)=13$, \Absolute minimum: $\displaystyle f\left(\pm\frac{1}{\sqrt{2}}\right)=\frac{3}{4}$}

\item $\displaystyle f(x)=x^3+5x^2+3x+1$ on $[-1,0]$.
\answer{\Absolute maximum: $\displaystyle f(-1)=2$, \Absolute minimum: $\displaystyle f\left(-1/3\right)=\frac{14}{27}$}

\item $\displaystyle g(t)=\sqrt{t}\left(t-2\right)$ on $[0,1]$.
\answer{\Absolute maximum: $\displaystyle g(0)=0$, \Absolute minimum: $\displaystyle g\left(2/3\right)=-\frac{4}{3}\sqrt{\frac{2}{3}}$}

\item $\displaystyle H(x)=x^{\frac{1}{3}}-3$ on $[-1,8]$.
\answer{\Absolute maximum: $\displaystyle H(8)=-1$, \Absolute minimum: $\displaystyle H\left(-1\right)=-4$}

\item $f(x)=\sin x\cos x$ on $[0,2\pi]$
\answer{\Absolute maximum: $\displaystyle f(\pi/4)=f(5\pi/4)=\frac{1}{2}$, \Absolute minimum: $f(3\pi/4)=f(7\pi/4)=-\displaystyle\frac{1}{2}$}


\end{problemblock}
