Absolute and Local Extrema

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Definitions

Intuitively, the absolute maximum value is the largest of the possible values of $f(x)$, and similarly for minimum. Notice that a function may reach its maximum (or minimum) at more than one $x$-value.
DO: Read the following definition carefully, and slowly, with an example $f$ graphed, and think of what this precise language is saying.

Definition (Absolute Extrema)

If $c$ is a number in the domain of $f$, then $f(c)$ is the absolute maximum value of $f$ if $f(c)\geq f(x)$ for all $x$ in the domain of $f$.
If $c$ is a number in the domain of $f$, then $f(c)$ is the absolute minimum value of $f$ if $f(c) \leq f(x)$ for all $x$ in the domain of $f$.

A value of $f(x)$ may not be the largest (or smallest) of all, but it might be the largest (or smallest) compared to nearby values. We call these local extrema, or local extreme values.
DO: Read the following definition carefully and look at an example or two.

Definition (Local Extrema)

If $c$ is a number in the domain of $f$, then $f(c)$ is a local maximum value of $f$ if $f(c) > f(x)$ when $x$ is "near" $c$.
Likewise, $f(c)$ is local minimum value of $f$ if $f(c) < f(x)$ when $x$ is "near" $c$.

Notice that when a function is defined on a closed interval, an absolute extreme value may occur at the endpoint of that interval of domain, since the endpoint of the interval may yield the largest value of $f$ on that closed interval.

This situation is different with local extrema. When we say $x$ is near $c$, we mean on an open interval containing $c$, so on either side of $c$. When $f$ is defined on a closed interval, there is no open interval containing an endpoint of the closed interval on which $f$ is defined. Hence, a local extreme value cannot occur at the endpoint of an interval of domain. This is a definition, and it could be defined differently.

This video will help clarify these concepts.

Notice that the absolute and local maxima and minima are $y$-values. Graphically, this means that the max/min value is the maximum/minimum height of the graph at some $x=c$. Then $x=c$ is where the max/min occurs.

Example: Suppose you are driving westward across Colorado on IH 70 (to get to Vail to go snowskiing). As you head west, the road will sometimes be going up, and other times going down, but overall your altitude is increasing.

Suppose that, along the way, you go up, then at mile marker 324 (where your altitude is 6,200 feet) you start going down. We would say that you reached a local maximum altitude of 6,200 feet at $x=324$. You keep driving, some up, some down (attaining local maximum and maximum altitudes) until you reach Vail Pass (the Continental Divide) at mile marker 278, where the altitude is 10,662 feet. We say you reached an absolute maximum altitude of 10,662 feet at $x=278$. If you are traveling from Denver at 5280 feet high, to Grand Junction at 4560 feet high, what is your guess about the absolute minimum altitude you attain, and where it occurs?

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Absolute and Local Extrema

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