Use the following graphs to identify the points (if any) on th...

Question

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value

Accepted Answer

Hello. In this video, we are going to be identifying the values of X, where we have an absolute maximum or minimum value. The problem states to examine the function F of X plotted on the interval from 1 to 5. We want to identify the X values where the function attains its absolute maximum or minimum value. Now, what does it mean to have an absolute maximum or minimum? What this means is that we want to identify the portions of the graph where it reaches its lowest or highest value on the defined interval. Let's go ahead and start by identifying the absolute minimum. Now, if we are taking a look at the given graph, what seems to be the lowest possible point that the graph reaches in the defined interval? Well, the lowest possible point is located at the cusp. Approximately, this has a value of 1.5, and this is located at the respective X value, X is equal to 3. Since this is the lowest possible Y value that the graph has on the defined interval, this point is going to be the absolute minimum, and the respective value of X where this is located at, is going to be X is equal to 3. So X is equal to 3 is going to be the location of the absolute minimum. Now what about the absolute max? While the absolute maximum value is the highest possible value that the graph reaches in the defined interval. Now, this one is a little bit tricky, but the absolute maximum is going to be located at the X value, X is equal to 1. Now, on the graph itself, it seems to have the value Y is equal to 2. But notice that this value on the graph is represented with an open circle. That means that at the value Y is equal to 2, this point is not included in the domain, but we do have another defined point located at Y is equal to 3. This defined point is also located on the respective X value X is equal to 1. So the highest defined value that the graph has on the interval is located at the Y value Y is equal to 3. And since this is located at X is equal to 1, that means the location of the absolute maximum will be X is equal to 1. And these are going to be the solutions to the problem. So I hope this video helps you in understanding on how to identify the absolute maximum or minimum, and I will go ahead and see you all in the next video.

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Key Concepts

Absolute Maximum and Minimum

Critical Points

Closed Interval

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