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%  Curve Sketching I                                    %
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\problemgrouptitle{Curve Sketching I}

\begin{problemblock}[3]{Find the domain, intercepts, intervals of increase and decrease, \relative maxima and minima, intervals of concave upward and downward, and inflection points of the given functions and then sketch their graphs.}
\item $\displaystyle f(x)=x^3-6x^2$ \answer{%
$D=\mathset{R}=(-\infty,\infty)$;
$x$-int$=0,\, 6$; $y$-int$=0$;
Increasing on: $(-\infty,0)\cup(4,\infty)$; Decreasing on: $(0,4)$;
\Relative maximum: $f(0)=0$; \Relative minimum: $f(4)=-32$;
Concave upward on: $(2,\infty)$; Concave downward on: $(-\infty,2)$;
Inflection point: $(2,-16)$; Graph:
\begin{center}
% Note that no comments are allowed when using \asycode
\resizebox{5cm}{!}{\asycode{%
import graph;
size(200,200,IgnoreAspect);
real f(real x) {return x^3-6*x^2;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-2.2,6.5,operator ..),red+2);
xaxis("$x$",xmax=7,Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4));
dot(F(0),4+black);
label("$(0,0)$",F(0),NW);
dot(F(2),4+black);
label("$(2,-16)$",F(2),NE);
dot(F(4),4+black);
label("$(4,-32)$",F(4),S);
xtick("$6$",6);
label("$f(x)=x^3-6x^2$",F(6.3),W);
}}
\end{center}
}

\lab{\addtoanswerkey{\clearpage}}
\item $\displaystyle g(t)=t^{\frac{3}{2}}-3t^{\frac{1}{2}}$\answer{%
$D=[0,\infty)$;
$t$-int$=0,\, 3$; $y$-int$=0$;
Increasing on: $(1,\infty)$; Decreasing on: $(0,1)$;
No~\relative maxima; \Relative minimum: $g(1)=-2$;
Concave upward on: $(0,\infty)$; Not concave downward anywhere;
No~inflection points; Graph:
\begin{center}
\resizebox{5cm}{!}{\asycode{%
import graph;
size(200,200);
real f(real x) {return x^(3/2)-3*x^(1/2);}
pair F(real x) {return (x,f(x));}
xaxis("$t$",Arrow(HookHead,size=4));
yaxis("$y$",Arrow(HookHead,size=4));
draw(graph(f,0,4,operator ..),red+2);
dot(F(1),4+black);
label("$(1,-2)$",F(1),S);
xtick("$3$",3);
label("$g(t)=t^{\frac{3}{2}}-3t^{\frac{1}{2}}$",F(3.5),W);
}}
\end{center}
}
\item $\displaystyle F(x)=\sqrt{x^2+9}$\answer{%
$D=\mathset{R}=(-\infty,\infty)$;
No~$x$-int; $y$-int$=3$;
Increasing on: $(0,\infty)$; Decreasing on: $(-\infty,0)$;
No~\relative maxima; \Relative minimum: $F(0)=3$;
Concave upward on: $(-\infty,\infty)$; Not concave downward anywhere;
No~inflection points; Graph:
\begin{center}
\resizebox{5cm}{!}{\asycode{%
import graph;
size(200,200);
real f(real x) {return sqrt(x^2+9);}
pair F(real x) {return (x,f(x));}
xaxis("$x$",xmax=3.5,Arrow(HookHead,size=4));
yaxis("$y$",Arrow(HookHead,size=4));
draw(graph(f,-3,3,operator ..),red+2);
dot(F(0),4+black);
label("$(0,3)$",F(0),SE);
label("$F(x)=\sqrt{x^2+9}$",F(3),NW);
}}
\end{center}
}
\end{problemblock}
