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Figure %: Graph of function
*f*
on the interval
[*a*, *e*]

If a function has an absolute maximum at
*x* = *b*
, then
*f* (*b*)
is the largest value
that
*f*
can attain. Similarly, if a function has an absolute minimum at
*x* = *b*
, then
*f* (*b*)
is the smallest value that
*f*
can attain.

On the graph above of the function
*f*
on the closed interval
[*a*, *e*]
, the point
(*a*, *f* (*a*))
represents the absolute minimum, and the point
(*d*, *f* (*d* ))
represents the absolute
maximum.

To define these terms more formally: a function
*f*
has an absolute maximum at
*x* = *b*
if
*f* (*b*)≥*f* (*x*)
for all
*x*
in the
domain of
*f*
. A function
*f*
has an absolute minimum at
*x* = *b*
if
*f* (*b*)≤*f* (*x*)
for all
*x*
in the domain of
*f*
. Together, the absolute minimum and the absolute maximum
are known as the absolute extrema of the function.

A function
*f*
has a local maximum at
*x* = *b*
if
*f* (*b*)
is the largest value that
*f*
attains "near
*b*
." Similarly, a function
*f*
has a local minimum at
*x* = *b*
if
*f* (*b*)
is the smallest value that
*f*
attains "near
*b*
." In the graph above,
*f*
has a local
maximum at
*x* = *b*
and at
*x* = *d*
, and has a local minimum at
*x* = *c*
.

Taken together, the local maxima and local minima are known as the local extrema. A local minimum or local maximum may also be called a relative minumum or relative maximum.

To define these terms more formally: a function
*f* (*x*)
has a local (or relative) maximum at
*x* = *b*
if there is an open interval
*I*
in which
*f* (*b*)≥*f* (*x*)
for all
*x*
in
*I*
. A function
*f* (*x*)
has a local (or relative)
minimum at
*x* = *b*
if there is an open interval
*I*
in which
*f* (*b*)≤*f* (*x*)
for all
*x*
in
*I*
.

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