3-4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is show

3-4 For each of the numbers a, b, c, d, r, and s, state...

Key Concepts

Absolute Extrema

An absolute maximum (or minimum) is the greatest (or least) value that a function achieves over its entire domain. This occurs when there is no other point in the domain where the function takes a higher (or lower) value. The identification of absolute extrema often involves checking both interior critical points and the endpoints of the domain if the function is defined on a closed interval.

Local (Relative) Extrema

A local maximum (or minimum) is a point where the function's value is higher (or lower) than all nearby points, even if it isn't the highest (or lowest) value overall. Local extrema are determined by examining the behavior of the function in a neighborhood around a point, and they might not qualify as absolute extrema when considering the entire domain of the function.

Critical Points

Critical points refer to points in the domain of a function where the derivative is zero or undefined. These points are important in the study of extrema because they are potential locations for local maximums or minimums. Identifying and analyzing critical points is a central part of determining where a function might have local or absolute extrema.

Endpoints and Domain Considerations

When a function is defined on a closed interval, the endpoints must be analyzed in addition to interior points because absolute extrema can occur at these boundary points even if the function's behavior in the immediate neighborhood does not indicate a local extremum. This concept emphasizes the importance of the domain's structure when interpreting the nature of any extrema.

Transcript

00:01 Hi everyone, so for today's problem, we're going to figure out whether these points given to us are absolute maximin, local maximin, or neither.

00:12 So let's first start by defining what a maximum or minimum is.

00:20 When we're looking at graphs, it's actually pretty easy to tell.

00:24 If you have a curve that goes upwards like this, the point right at the bottom is considered a minimum.

00:35 If you have a curve that goes like this, the point at the top is considered a maximum.

00:45 Great.

00:47 So looking towards our graph, i like to think of them as valleys and hills.

00:53 So you have your maxes at your hills and then you have your minns at the valleys.

00:59 But in cases where you can't have a min or max, let's write that down.

01:09 For example, if you have a point that doesn't exist.

01:12 So you guys may know this, you have a continuous little line, and then you have a point that doesn't exist, and then you keep going.

01:18 So at that point, there can't be a local man or max because it doesn't exist.

01:25 Another example is if you have a cusp.

01:29 I think of it as like a bird drawing, but they are cusps, and at this point, it's way too sharp.

01:39 You have to have a rounded hill like these ones, in order to.

01:43 To have a maxerman.

01:46 And then lastly, we have jump.

01:51 It kind of starts out like this, but then you jump up like this.

01:58 So if you have a jump, the maxerman can't exist here either.

02:04 So in all three of these cases, maxerman does not exist.

02:12 Okay, now that we've clarified that, let's start by finding our local maxermins.

02:18 And then let's begin at point a.

02:22 And i'd like to draw out the part of the graph that got cut off.

02:28 So i imagine it would look like this.

02:31 So that means point a exists at the top of a rounded hill.

02:37 And then we know that the point that exists at a top of a rounded hill is a maximum.

02:42 So point a is a local maximum.

02:46 Let's write that down.

02:49 There we go.

02:50 Our first point.

02:51 All right, let's keep going.

02:53 And we go down our slope and we reach point.

02:55 B, but point b is too sharp and pointy.

03:01 It's definitely not a rounded tail...

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