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As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum.
An extremum (or extreme value) of a function is a point at which a maximum or minimum value of the function is obtained in some interval. A local extremum (or relative extremum) of a function is the point at which a maximum or minimum value of the function in some open interval containing the point is obtained.
An extremum (plural: extrema) is a maximum (plural: maxima) or a minimum. Let $f$ be a function on $I\subset\mathbb{R}$ and let $x_0\in\ I$. We say that $f$ have a local maximum (resp. local minimum) at $x_0$ if $f(x)<f(x_0)$ (resp. $f(x)>f(x_0)$) in the strict neighborhood of $x_0$ (i.e. $|x-x_0|<\epsilon$ for some $\epsilon>0$ and $x\not=x_0$).
When we are trying to find the absolute extrema of a function on an open interval, we cannot use the Extreme Value Theorem. However, if the function is continuous on the interval, many of the same ideas apply. In particular, if an absolute extremum exists, it must also be a relative extremum.
Explain how to find the critical points of a function over a closed interval. Describe how to use critical points to locate absolute extrema over a closed interval. Given a particular function, we are often interested in determining the largest and smallest values of the function.
Find all critical points of f f that lie over the interval (a,b) (a, b) and evaluate f f at those critical points. Compare all values found in (1) and (2). From the location of absolute extrema, the absolute extrema must occur at endpoints or critical points.
Extrema are the extreme values of a function - the places where it reaches its minimum and maximum values. That is, extrema are the points of a function where it is the largest and the smallet. We can identify two types of extrema - local and global.
