Can a relative maximum be an absolute maximum?

Can a Relative Maximum Really Be the Absolute Best? Let’s Break It Down.

Calculus. It can sound intimidating, right? But beneath the jargon, there are some really cool ideas. One of those is understanding the difference—and the connection—between relative and absolute maximums. It’s a question that pops up all the time: Can a relative maximum also be the absolute maximum? Short answer? Absolutely!

But what do these terms even mean? Let’s unpack them.

Think of a “relative maximum” as a peak on a graph, but only compared to its immediate neighbors. It’s like the highest point in a small valley. The official definition? A function f(x) has a relative maximum at a point x = c if f(c) is bigger than or equal to all the f(x) values right around c. Easy enough.

Now, an “absolute maximum” is the king of the hill—the highest point anywhere on the graph. Formally, f(x) has an absolute maximum at x = c if f(c) is bigger than or equal to every single f(x) value in the entire domain. It’s the ultimate high point.

So, here’s the kicker: what happens when that “small valley” peak is also the highest point overall? Bingo! The relative maximum is the absolute maximum. It’s like finding out that the tallest hill in your backyard is actually the tallest mountain in the world (okay, maybe not that likely, but you get the idea).

Let’s Make It Real

Imagine hiking in the mountains. You climb to the top of a hill, feeling pretty good. You’re at a relative maximum—you’re higher than everything around you. But then you look around and realize that this hill is actually part of a much larger mountain range, and there’s a peak way off in the distance that’s much, much higher. Your hill was just a relative maximum. But what if, when you got to the top of your hill, you realized there wasn’t anything higher? You’d be at both a relative and an absolute maximum!

Or, think about the function f(x) = -x2. It’s a simple parabola that opens downwards. The very top of that parabola, at x = 0, is a relative maximum. But guess what? It’s also the absolute maximum because nothing on that curve ever goes higher than that point.

A Few Things to Keep in Mind

• A function can have lots of “local” peaks (relative maxima), but there’s only ever one true “highest point” (absolute maximum).
• That absolute maximum might be hiding at the very edge of the graph, or at a critical point somewhere in the middle.
• If you’re dealing with a nice, continuous function on a closed interval, you’re guaranteed to have both an absolute high point and an absolute low point. That’s the Extreme Value Theorem in action.

The Bottom Line

So, can a relative maximum be an absolute maximum? You bet! It just means that the highest point in a specific area also happens to be the highest point anywhere on the function’s graph. It’s a simple concept, but it’s crucial for solving all sorts of problems, from figuring out how to maximize profits to designing the most efficient bridge. And that, my friends, is why understanding calculus can actually be pretty useful in the real world.