\begin{homework}\
\label{homework:13}

\medskip

\exercise %1.
Suppose that $n$ is an integer greater than 2.  On the curve $y=f(x)=x^n$,
what sort of point is the origin?  Sketch the curve, and indicate the
concavity.

\smallskip

\exercise %2.
In parts (a)-(j) below, find each max/min point (using the second
derivative test to be sure what type of point it is), point of inflection,
and asymptote (vertical, horizontal, or slanted), if there are any.  Also
indicate where the curve is concave up and concave down.  Sketch the graph.

\noindent
(a) $y=x^2-x$, (b) $y=2+3x-x^3$, (c) $y=x^3-9x^2+24x$,
(d) $y=x^4-2x^2+3$, (e) $y=3x^4-4x^3$,
(f) $y=(x^2-1)/x$,
(g) $y=3x^2-(1/x^2)$, (h) $y=\cos(2x)-x$, (i) $y=\sin x+\cos x$,
(j) $y=\tan(x/2)-x$.

\smallskip

\exercise %3.
Suppose  $\begin{cases} f'(x) > 0& \text{for $|x| > 2$;}\\
                                  f'(x) < 0& \text{for $|x| < 2$;}\\
                                  f''(x) < 0& \text{for $x  < 0$; and }\\
                                  f''(x) > 0& \text{for  $x  > 0$}\end{cases}$.

\noindent
From the given information, sketch a possible graph of $f(x)$.
How could your answer vary and still be correct?


\smallskip

\exercise %4.
Suppose $f''(x) <0$ for $x<1$ and  $f''(x)>0$ for $x>1$.
Suppose also that $f(1)=1$.  Sketch a possible graph of $f(x)$, assuming that

         i) $f'(1) = 0$,   ii) $f'(1) > 0$ and  iii) $f'(1) < 0$.

\smallskip

\begin{rightindent}{3.4in}
\exercise %5.
At the right is a sketch of $y=x^6-5x^4$.

(a) Find the exact coordinates $(x,y)$ of the local maxima and local
minima.

(b) For what values of $x$ is the function concave upward?
									
(c) Find the exact $x$-coordinates of all points of inflection. 
({\bf Hint:} Color the part of the curve that is concave up blue and color
the part that is concave down red. Points of inflection occur {\it only}
where the curve {\it changes} color!)
\end{rightindent}
\hfill
\begin{psfigure}{2.0in}{2.0in}{figNJ.ps}
\end{psfigure}

(d) Explain how you {\it know from your calculations} (not from the
sketch you were given) which of your answers to part (a) are minima and
which are maxima.


\end{homework}

