How to Find a Function’s Absolute Extrema over Its Entire Domain
A function’s absolute max and absolute min over its entire domain are the single highest and single lowest values of the function anywhere it’s defined. A function can have an absolute max or min or both or neither. For example, the parabola
has an absolute min at the point (0, 0) — the bottom of its cup shape — but no absolute max because it goes up forever to the left and the right. You could say that its absolute max is infinity if it weren’t for the fact that infinity is not a number and thus it doesn’t qualify as a maximum — and ditto, of course, for negative infinity as a minimum.
The basic idea is this: Either a function will max out somewhere or it will go up forever to infinity. And the same idea applies to a min and going down to negative infinity.
To locate a function’s absolute max and min over its domain, first find the height of the function at each of its critical numbers. And since you’re looking for absolute extrema, you consider all the critical numbers, not just those in a closed interval. The highest of these values is the absolute max unless the function rises to positive infinity somewhere, in which case you say that it has no absolute max. The lowest of these values is the absolute min, unless the function goes down to negative infinity, in which case it has no absolute min.
If a function goes up to positive infinity or down to negative infinity, it does so at its extreme right or left or at a vertical asymptote. So, your next step (after evaluating all the critical points) is to evaluate
This is the so-called end behavior of the function. Finally, if the function has any vertical asymptotes, you evaluate the limit of the function as x approaches each asymptote from the left and from the right. If any of the above limits equals positive infinity, then the function has no absolute max; if none equals positive infinity, then the absolute max is the function value at the highest of the critical points. And if any of the limits is negative infinity, then the function has no absolute min; if none of them equals negative infinity, then the absolute min is the function value at the lowest of the critical points.
The figure shows a couple functions where the above method won’t work. The function f(x) has no absolute max despite the fact that it doesn’t go up to infinity. Its max isn’t 4 because it never gets to 4, and its max can’t be anything less than 4, like 3.999, because it gets higher than that, say 3.9999. The function g(x) has no absolute min despite the fact that it doesn’t go down to negative infinity. Going left, g(x) crawls along the horizontal asymptote at y = 0, but it never gets as low as zero, so neither zero nor any other number can be the absolute min.