%% Math 131, Fall, 2003
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\noindent\textbf{Name}:\ \vbox{\hrule width 3in}

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\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Quiz \#9 Solutions} \\
Professor Hwang \\
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\be

\item Let $f(x)=x^3-12x+3$. Find the critical and inflection point(s)
  of~$f$, and classify each critical point as a local maximum, local
  minimum, or neither.

  \soln Differentiating gives $f'(x)=3x^2-12$ and $f''(x)=6x$. The
  critical points are solutions of $3x^2-12=0$, or $x=\pm2$. The second
  derivative is zero when $x=0$. The sign diagrams are:
  \[
  \input{qsoln09a.eepic}\qquad\qquad\input{qsoln09b.eepic}
  \]
  We see immediately that $x=-2$ is a local maximum, $x=2$ is a local
  minimum, and that $x=0$ is an inflection point.

  \sep

\item Find the absolute maximum and absolute minimum values of
  $f(x)=x^3-12x+3$ on the interval $[-1,3]$. (Note: This is the same
  function as in the previous question.)

  \soln The maximum and minimum values must occur at either a critical
  point of~$f$ or at an endpoint of~$[-1,3]$. The only critical point
  of~$f$ in the interval is $x=2$. Direct computation gives
  \[
  f(-1)=(-1)+12+3=14,\qquad
  f(2)=8-24+3=-13,\qquad
  f(3)=27-36+3=-6.
  \]
  The absolute maximum is~$14$ (at $x=-1$) and the absolute minimum
  is~$-13$ (at $x=2$).

\ee
\eject

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