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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{The Mean Value Theorem}

\begin{example}
Show that the function \( f(x)=x^{5}+4x^{3}+3x-8 \) has only one root, one
point \( x \) where \( f(x)=0 \).
\end{example}

\subsubsection*{Solution}

If \( f(x) \) had two points, call them \( a \) and \( b \), \( a\neq b \),
with \( f(a)=0 \) and \( f(b)=0 \), then there would have to be, by the MVT
(actually, by Rolle's Theorem) a point \( c \) between \( a \) and \( b \)
for which \( f'(c)=0 \). However, \[
f'(x)=5x^{4}+4x^{2}+3,\]
so no matter what \( c \) is, \( f'(c)\neq 0 \), in fact, \( f'(c)\geq 3 \)
at every point. Since this can't exist, the assumption, that there were two
distinct roots of the equation, had to have been wrong.
\end{document}
