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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Curve Sketching}

\begin{thm}
\textbf{\emph{{[}The Second Derivative Test{]}}}. If \( f(x) \) has a critical
point at \( c \), then:
\begin{enumerate}
\item If \( f''(c)>0 \), then \( f(x) \) has a relative minimum at \( c \),
\item If \( f''(c)<0 \), then \( f(x) \) has a relative maximum at \( c \), and 
\item If \( f''(c)=0 \), then you can't tell whether \( f(x) \) has a relative maximum,
or minimum, or neither, at \( c \).
\end{enumerate}
\end{thm}
\textbf{\emph{Proof.}}

This actually reduces to the first derivative test, since \( f''(c)>0 \) means
that \( f'(x) \) is \textit{increasing} at \( c \), so that \( f'(x) \) (which
is 0 at \( c \)) must go from negative to positive as you cross \( c \), which
describes a minimum. The idea for the second case is the same. For the last,
you have to think about examples, to see that any of the possibilities might
occur. Examples of this {}``problem{}'' are:

\begin{enumerate}
\item \( f(x)=x^{4} \). \( f'(0)=0 \), and \( f''(0)=0 \), but this is a local
minimum,

\begin{enumerate}
\item \( f(x)=-x^{4} \). \( f'(0)=0 \), and \( f''(0)=0 \), but this is a local
maximum,
\item \( f(x)=x^{3} \). \( f'(0)=0 \), and \( f''(0)=0 \), but this is neither
a local minimum, no a local maximum.\end{enumerate}
\end{enumerate}

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