In most applications, the domain of the functions involved are restricted to numbers in a particular interval. Intuitively, an absolute maximum is the highest point \((c,f(c))\) on the graph of \(f\) over the interval \([a,b]\text{.}\) Similarly, an absolute minimum is the lowest point on the graph over \([a,b]\text{.}\)
Definition4.3.1.
Let \(f\) be a function defined on \([a,b]\text{,}\)\(c \in [a,b]\text{.}\) Then,
\(f\) has an absolute maximum on \([a,b]\) at \(x = c\text{,}\) if \(f(x) \leq f(c)\) for all \(x \in [a,b]\text{.}\)
\(f\) has an absolute minimum on \([a,b]\) at \(x = c\text{,}\) if \(f(x) \geq f(c)\) for all \(x \in [a,b]\text{.}\)
Subsection4.3.1Extreme-Value Theorem
Theorem4.3.2.
Let \(f\) be a function, continuous on \([a,b]\text{.}\) Then, \(f\) has both an absolute maximum and absolute minimum on \([a,b]\text{.}\) More explicitly, there exists \(c_1, c_2 \in [a,b]\) such that \(f(c_1) \leq f(x) \leq f(c_2)\) for all \(x \in [a,b]\text{.}\)
The extreme value theorem is intuitively clear, however, a proof of the extreme value theorem requires more advanced mathematical analysis.
Subsection4.3.2Absolute Extrema Test
Recall that if \(f\) has a maximum value on an interval \([a,b]\text{,}\) then it can only occur at a critical point of \(f\) (that is, where \(f'(x) = 0\) or \(f'(x)\) does not exist but \(f\) is defined). On a closed interval, a function can also have extrema at its endpoints \(x = a\) or \(x = b\text{.}\) In summary,
Theorem4.3.3.Characterization of absolute extrema.
Let \(f\) be continuous on \([a,b]\text{.}\) Suppose that \(f\) has \(n\) critical points on \([a,b]\text{.}\) Then, \(M\) be the maximum of the \(n+2\) values of \(f\text{,}\) evaluated at the \(n\) critical points, along with the endpoints of the interval. Then, \(M\) is the maximum value of \(f\) on \([a,b]\text{.}\) Similarly, the minimum \(m\) of these values is the minimum value of \(f\) at \([a,b]\text{.}\)
If \(f\) has only one critical point, then it isn't necessary to consider the endpoints. For example, if \(f'\) changes sign from positive to negative at \(x = c\text{,}\) then \(f\) is increasing on \((a,c)\) and decreasing on \((b,c)\text{,}\) so the maximum occurs at \(x = c\text{.}\)
Subsection4.3.3Absolute Extrema Over Infinite Intervals
Absolute extrema can also be consider over infinite intervals, and arbitrary domains.
Definition4.3.4.
Let \(f\) be a function, \(c \in D(f)\text{.}\) Then,
\(f\) has an absolute maximum at \(x = c\text{,}\) if \(f(x) \leq f(c)\) for all \(x\) in the domain of \(f\text{.}\)
\(f\) has an absolute minimum at \(x = c\text{,}\) if \(f(x) \geq f(c)\) for all \(x\) in the domain of \(f\text{.}\)
Subsection4.3.4Remark on Absolute Extrema
In general, absolute extrema are also considered local extrema, by default. However, if you are using the convention that endpoints cannot be local extrema, then it is possible for an endpoint to be an absolute extrema, while not being a local extrema.
