page 1 of 3

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*.

Take a Study Break!