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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Mean Value Theorem">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 21:21:17 1997}
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\section{The Mean Value Theorem\label{Mean value theorem}}

\subsubsection{\QTR{BOXHEAD}{Rolle's Theorem\label{Rolle's theorem}}}

Let $f$ be a function that satisfies the following three hypotheses:

\begin{enumerate}
\item[1.]  $f$ is continuous on the closed interval $\lbrack a,b\rbrack $.

\item[2.]  $f$ is differentiable on the open interval $(a,b)$.

\item[3.]  $f(a)=f(b)$
\end{enumerate}

Then there is a number $c$ in $(a,b)$ such that $f^{\prime }(c)=0$.

\subsubsection{\QTR{BOXHEAD}{Mean Value Theorem}}

Let $f$ be a function that satisfies the following hypotheses:

\begin{enumerate}
\item[1.]  $f$ is continuous on the closed interval $\lbrack a,b\rbrack $.

\item[2.]  $f$ is differentiable on the open interval $(a,b)$.
\end{enumerate}

Then there is a number $c$ in $(a,b)$ such that 
\[
f^{\prime }(c)=\frac{f(b)-f(a)}{b-a} 
\]
or, equivalently, 
\[
f(b)-f(a)=f^{\prime }(c)(b-a) 
\]

\subsubsection{Constant Functions}

\begin{theorem}
If $f^{\prime }(x)=0$ for all $x$ in an interval $(a,b)$, then $f$ is
constant on $(a,b)$.
\end{theorem}

\subsubsection{\QTR{BOXHEAD}{Equal Derivatives}}

\begin{theorem}
If $f^{\prime }(x)=g^{\prime }(x)$ for all $x$ in an interval $(a,b)$, then $%
f-g$ is constant on $(a,b)$; that is, $f(x)=g(x)+c$ where $c$ is a constant.
\end{theorem}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Computing a derivative}{}{}{../../Help/DM7-3.tex#Derivative%
}}

{\small \hyperref{Derivatives of piecewise-defined functions}{}{}{%
../../Help/DM7-3.tex#Piecewise-defined function}}

{\small \hyperref{Optimization}{}{}{../../Help/DM7-3.tex#Optimization}}

{\small \hyperref{Plotting derivatives}{}{}{../../Help/DM7-3.tex#Plotting
Derivatives}}

\noindent \hrulefill

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