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 point. More precisely,
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'll call it a local extrema. More precisely,
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 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 $x=c$.
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?