on April 22, 2022

Can you have absolute extrema on an open interval?

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Absolute Extrema on Open Intervals: A Real-World Look

So, you’re diving into calculus and trying to wrap your head around finding the highest and lowest points of a function, huh? We call those extrema, and they can be either absolute (the ultimate high or low) or relative (just a local peak or valley). Finding these points on closed intervals is pretty straightforward, but what about open intervals? That’s where things get a little trickier. Let’s break it down.

What Exactly Are Absolute Extrema?

First, let’s make sure we’re all on the same page. An absolute maximum is simply the highest point a function reaches on a given interval. Think of it like the summit of a mountain range. Conversely, the absolute minimum is the lowest point – the bottom of the deepest valley.

Now, there’s this thing called the Extreme Value Theorem. It’s a big deal because it basically guarantees that if you have a continuous function on a closed interval (meaning it includes the endpoints), you will find both an absolute max and an absolute min. It’s like a safety net for your calculations. But here’s the catch: this theorem only works for closed intervals. So, what happens when we open things up?

The Open Interval Conundrum

An open interval, in math-speak, is one that doesn’t include its endpoints. We’re talking about something like (a, b), where you get super close to ‘a’ and ‘b’, but you never actually reach them. This tiny difference can throw a wrench into our plans for finding absolute extrema.

Why? Well, imagine trying to find the highest point on a hill if you’re not allowed to stand at the very top. That’s essentially what we’re dealing with. Take the function f(x) = x on the interval (0, 1). As x gets closer and closer to 1, f(x) also gets closer to 1, but it never actually gets there. So, there’s no true “highest point,” no absolute maximum. The same goes for the minimum as x approaches 0. It’s like chasing a ghost.

When Can You Actually Find Absolute Extrema on Open Intervals?

Okay, it’s not all doom and gloom. You can find absolute extrema on open intervals, but you need the right conditions. Think of it like finding a hidden treasure – you need the right map.

  • The Lone Critical Point Scenario: Imagine a rollercoaster with only one hill or one valley. If your function on an open interval has only one critical point (where the slope is zero or undefined), then things get interesting:

    • If that point is a local maximum (the top of a hill), guess what? It’s also the absolute maximum.
    • And if that point is a local minimum (the bottom of a valley), it’s the absolute minimum.

    Think of a simple parabola opening upwards. Its vertex is the lowest point, and on an open interval, it’s the lowest point.

  • The Endpoint Limit Trick: Even if you don’t have a critical point, you might still find absolute extrema if the function sort of “settles down” to a specific value as you approach the endpoints. If those values are the highest or lowest the function reaches, bingo!

    But be careful! If the function goes wild and heads towards infinity (or negative infinity) as you approach an endpoint, then you’re out of luck. No absolute max or min for you.

    • When to Give Up the Search

    It’s just as important to know when absolute extrema aren’t going to show up. Save yourself some time and frustration by watching out for these situations:

  • The Unbounded Function Frenzy: If your function is like a rocket ship, shooting off towards infinity or plummeting towards negative infinity, you won’t find both an absolute max and an absolute min. Think of f(x) = 1/x on the interval (0, 1). It just keeps getting bigger and bigger as you get closer to zero. No top in sight!
  • The Discontinuity Disaster: If your function has a break or a jump in it (it’s discontinuous), then the Extreme Value Theorem is out the window. And that means no guarantee of absolute extrema.

    • How to Hunt for Absolute Extrema on Open Intervals: A Step-by-Step Guide

    Alright, ready to put on your calculus detective hat? Here’s how to track down those elusive absolute extrema:

  • Find Those Critical Points: First, find where the derivative of your function is zero or undefined within the open interval. These are your prime suspects.
  • Check the Neighborhood: If there is only one critical point, use the second derivative test to determine if it’s a local maximum or minimum. If f”(c) > 0, then f(c) is a local minimum, and if f”(c) < 0, then f(c) is a local maximum.
  • Stalk the Endpoints: Take a peek at what happens to your function as you get closer and closer to the endpoints of the open interval. Are the values settling down, or are they going haywire?
  • Consider Limits: If the limit of the function as it approaches an endpoint is greater than every relative maximum, there is no absolute maximum; if the limit is less than every relative minimum, there is no absolute minimum.
  • Piece It All Together: Now, put all the clues together – the critical points, the end behavior, and whether your function is well-behaved. Can you confidently say you’ve found the highest and lowest points?

    • The Bottom Line

    So, can you find absolute extrema on open intervals? The answer is a resounding “maybe!” While the Extreme Value Theorem is your best friend on closed intervals, you need to be a bit more of a detective on open intervals. Keep an eye out for those critical points, pay attention to the endpoints, and remember that not all functions play nice. With a little practice, you’ll be spotting absolute extrema like a pro!

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