We know how to find the extreme values of a continuous...

We know how to find the extreme values of a continuous function $f(x)$ by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.

Key Concepts

Domain Properties and Continuity

The existence of extreme values in continuous functions is strongly tied to the nature of their domain. If a continuous function is defined on an open or unbounded set, the absence of endpoints can mean that the function does not attain a maximum or minimum, even though it may be bounded in some sense.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function defined on a closed and bounded interval will have both a maximum and minimum value. However, this theorem does not apply to functions on open or unbounded intervals, where extreme values might not exist despite the function being continuous.

Critical Points

Critical points are locations in the domain of a function where its derivative is zero or undefined, potentially indicating a local maximum or minimum. The analysis of critical points is a standard technique for finding extreme values on a function's interior, but their absence can signal that the function does not have any interior extrema or that the extrema, if they exist, might be found at the domain’s boundaries.

Endpoints and Domain Boundaries

Endpoints refer to the boundary values of the function's domain, which are examined when applying the Extreme Value Theorem on a closed interval. In situations where a function lacks such endpoints—such as when defined on an open or infinite interval—the function may not achieve absolute maximum or minimum values even though its behavior might approach particular limits.

Transcript

00:01 Let's review what we know about finding extreme values of functions.

00:05 We know that if we have a closed continuous function, we need to look at the end points and critical points to find any minimum or maximum places on my graph.

00:26 As a reminder, critical points happen at two places, either when the derivative of my function equals zero, or when the derivative of my function is undefined.

00:41 So three places to check endpoints and critical points, which could be either zero or undefined.

00:50 So call those 2a and 2b.

00:51 Those are the points that i'm going to be looking at.

00:54 If i have a function that is not closed, i obviously don't have endpoints, so i'm just going to look at the critical points.

01:02 What if there are no critical points or end points? is it possible to even have that kind of a function? the answer is yes.

01:12 And there are some very simple functions that fall into that category.

01:15 For example, a straight line.

01:19 The line y equal x.

01:21 If you want to plot that out, and i'll just do it in black here, the line y equals x goes right through the origin.

01:28 We're going to pretend that i'm a better artist and it went right through the origin.

01:32 There are no end points.

01:34 I didn't specify a starting and ending point.

01:36 So it's going to go without bounds...

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