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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Maximum and Minimum Values">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 23:07:05 1997}
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\section{Maximum and Minimum Values\label{Maximum values}\label%
{Minimum values}}

\subsubsection{Maximum and Minimum Values}

\begin{enumerate}
\item  A function $f$ has an \textbf{absolute maximum} at $c$ if $f(c)\geq
f(x)$ for all $x$ in $D$, where $D$ is the domain of $f$. The number $f(c)$
is called the \textbf{maximum value} of $f$ on $D$. Similarly, $f$ has an 
\textbf{absolute minimum} at $c$ if $f(c)\leq f(x)$ for all $x$ in $D$ and
the number $f(c)$ is called the \textbf{minimum value} of $f$ on $D$. The
maximum and minimum values of $f$ are called the \textbf{extreme values} of $%
f$.

\item  A function $f$ has a \textbf{local maximum} (or \textbf{relative
maximum}) at $c$ if there is an open interval $I$ containing $c$ such that $%
f(c)\geq f(x)$ for all $x$ in $I$. Similarly, $f$ has a\textbf{\ local} 
\textbf{minimum }at $c$ if there is an open interval $I$ containing $c$ such
that $f(c)\leq f(x)$ for all $x$ in $I$.
\end{enumerate}

\begin{theorem}
(\QTR{BOXHEAD}{Extreme Value Theorem}) If $f$ is continuous on a closed
interval $\lbrack a,b\rbrack $, then $f$ attains an absolute maximum value $%
f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $%
\lbrack a,b\rbrack $.
\end{theorem}

\begin{theorem}
(\QTR{BOXHEAD}{Fermat's Theorem}) If $f$ has a local extremum (that is,
maximum or minimum) at $c$, and if $f^{\prime }(c)$ exists, then $f^{\prime
}(c)=0$.
\end{theorem}

\label{Critical number}A \textbf{critical number} of a function $f$ is a
number $c$ in the domain of $f$ such that either $f^{\prime }(c)=0$ or $%
f^{\prime }(c)$ does not exist.

\begin{theorem}
If $f$ has a local extremum at $c$, then $c$ is a critical number of $f$.
\end{theorem}

\subsubsection{Finding Absolute Maximum and Minimum Values}

To find the \textit{absolute} maximum and minimum values of a continuous
function $f$ on a closed interval $[a,b]$:

\begin{enumerate}
\item[1.]  Find the values of $f$ at the critical numbers of $f$ in $(a,b)$.

\item[2.]  Find the values of $f(a)$ and $f(b)$.

\item[3.]  The largest of the values from steps 1 and 2 is the absolute
maximum value; the smallest of these values is the absolute minimum value.
\end{enumerate}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Calculus, find extrema}{}{}{../../Help/DM7-3.tex#Find
extrema}}

{\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}}

{\small \hyperref{Solving exponential and logarithmic equations}{}{}{%
../../Help/DM3-5.tex#Solve, exponential equations}}

{\small \hyperref{Solving polynomial and rational equations}{}{}{%
../../Help/DM3-3.tex#Exact solutions}}

{\small \hyperref{Solving trigonometric equations}{}{}{%
../../Help/DM4-1.tex#Solve, trigonometry}}

{\small \hyperref{Solving equations numerically}{}{}{%
../../Help/DM3-3.tex#Numeric, solve}}

\noindent \hrulefill \hrulefill

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