How to Find Absolute Extrema over a Function’s Entire Domain

By Mark Ryan

A function’s absolute max and absolute min over its entire domain are the highest and lowest values (heights) of the function anywhere it’s defined. When you consider a function’s entire domain, a function can have an absolute max or min or both or neither. For example, the parabola y = x2 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 might think that its absolute max would be infinity, but infinity is not a number and thus it doesn’t qualify as a maximum (ditto for using negative infinity as an absolute min).

On the one hand, the idea of a function’s very highest point and very lowest point seems pretty simple, doesn’t it? But there’s a wrench in the works. The wrench is the category of things that dont qualify as maxes or mins.

Two functions with no absolute extrema.
Two functions with no absolute extrema.

In the figure, there are empty “endpoints” like (3,4) on f (x). f (x) doesn’t have an absolute max. 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. Similarly, an infinitesimal hole in a function can’t qualify as a max or min. For example, consider the absolute value function,

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you know, the V-shaped function with the sharp corner at the origin. It has no absolute max because it goes up to infinity. Its absolute min is zero (at (0, 0) of course). But now, say you alter the function slightly by plucking out the point at (0, 0) and leaving an infinitesimal hole there. Now the function has no absolute minimum.

Now consider g (x) in the figure. It shows another type of situation that doesn’t qualify as a min (or max). g (x) has no absolute min. Going left, g crawls along the horizontal asymptote at y = 0, always getting lower and lower, but never getting as low as zero. Since it never gets to zero, zero can’t be the absolute min, and there can’t be any other absolute min (like, say, 0.0001) because at some point way to the left, g will get below any small number you can name.

Keeping this in mind, here’s a step-by-step approach for locating a function’s absolute maximum and minimum (if there are any):

  1. Find the height of the function at each of its critical numbers. (Recall that a function’s critical numbers are the x-values within the function’s domain where the derivative is zero or undefined.)

    Consider all the critical numbers, not just those in a given interval. The highest of these values will be the function’s absolute max unless the function goes higher than that point in which case the function won’t have an absolute max. The lowest of those values will be the function’s absolute min unless the function goes lower than that point in which case it won’t have an absolute min. Steps 2 and 3 will help you figure out whether the function goes higher than the highest critical point and/or lower than the lowest critical point. If you apply Step 1 to g (x) in the figure, you’ll find that it has no critical points. When this happens, you’re done. The function has neither an absolute max nor an absolute min.

  2. Check whether the function goes up to infinity and/or down to negative infinity.

    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, evaluate

    image2.png

    — the so-called end behavior of the function — and the limit of the function as x approaches each vertical asymptote (if there are any) from the left and from the right. If the function goes up to infinity, it has no absolute max; if it goes down to negative infinity, it has no absolute min.

  3. Graph the function to check for horizontal asymptotes and weird features like the jump discontinuity in f (x) in the figure.

    Look at the graph of the function. If you see that the function gets higher than the highest of its critical points, it has no absolute max; if it goes lower than the lowest of its critical points, it has no absolute min. Applying this 3-step process to f (x) in the figure, Step 1 would reveal two critical points: the endpoint at (3, 1) and the local max at roughly (4.1, 1.3). In Step 2, you would find that f goes down to negative infinity and thus has no absolute min. Finally, in Step 3, you’d see that f goes higher than the higher of the critical points, (4.1, 1.3), and that it, therefore, has no absolute max. You’re done!