\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose $5x^3y -3xy^2 +y^3=6\, .$
$(1,2)$ is a point on this curve. Is the curve concave up or
concave down at $(1,2)$?

\hbox{\hsize=2.7in \parindent=0in
\vtop{{\bf Explicit way to go} $y$ {\it can} be solved as a function
of $x$.* Then you can differentiate the formula twice and evaluate
when $x=1$.
}
\hskip .2in
\vtop{{\bf Implicit way to go} Find $\displaystyle{{dy}\over {dx}}$ implicitly and then
differentiate again to get $\displaystyle{{d^2y}\over {dx^2}}$. Evaluate
everything at $(1,2)$.
}}

\vfootnote*{Here it is (really!):

$\hskip .15in\displaystyle{y = \left(-{5\over 2}x^4+3+x^3+ {1\over
{18}}\sqrt{1500x^9-675x^8-4860x^4+2916+1944x^3}\right)^{1/3} }$

\hskip .3in $\displaystyle{- \ {{\textstyle {\textstyle 5\vphantom{j^2}\over
\textstyle \vphantom{^2}3} x^3 -x^2} \over {\textstyle
\left(-{\textstyle\vphantom{j} 5\over \textstyle\vphantom{^2} 2}x^4+3+x^3+
{\textstyle 1\vphantom{j}\over \textstyle\vphantom{^2}
{18}}\sqrt{1500x^9-675x^8-4860x^4+2916+1944x^3}\right)^{1/3}}}+x}$}


\vfil\eject\end

