Identifying Extrema

In Exerc...

Question

Identifying Extrema

In Exercises 15–18:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

Graph of a function showing local maxima and minima, with labeled axes indicating increasing and decreasing intervals.

Accepted Answer

Welcome back, everyone. In this problem, we want to analyze the given graph of the function FF X on the interval open bracket -7, 11 closed bracket to find the intervals where the function is increasing or decreasing, and then identify its local and absolute extreme values. And here we have our graph of FXX. Now, let's break down this problem one by one. First, let's try to figure out where our graph is increasing. We can tell where it's increasing when we notice that uh it's, it's slope is going upwards. OK? Now if we think about it here, where is our graph increasing? We notice that if I were to highlight that in red, it's increasing on this interval. On this interval here in red and again on this interval, OK? So what are the values for our intervals here? Well, here we can see if our FX is increasing. A fix is increasing. On Here we have our first interval from -7 to -2, that is its X values. So increasing on the interval from -7 to -2, open closed parentheses. Union with our interval from 2 to 5. Again, open closed parentheses. Union with our interval from 6 to, well, here from 5 to 11, OK. So open parentheses from 5 to 11. So it's because it's increasing on those three intervals. On the other hand, On what interval is FFX decreasing? Well, I clear these. Notice that FF X is decreasing on this interval. OK, where the value is decreasing, its slope is going downwards. And now, this goes from -2 to 2. So FF X is decreasing on the interval, open parenthesis -22 closed parentheses. Now let's move to the next part of our problem. What is our function's local extreme values? Well, notice that the function changes from decreasing to increasing at X equals 2. So thus there is a local minimum at X equals 2. And notice that this minimum value is 6. OK. So now, that means we can say. Or local minimum. OK. Is F of 2 equals -6 or X value is 2, and our minimum value is -6. On the other hand, what is our local maximum going to be? Well, if we take a look here, OK, let me clear this off. If we take a look, that means we're looking at the point where here we can find our maximum value, OK? And no, notice that the function changes from increasing to decreasing at X equals -2. OK. So it does so right here and at X equals 5 because it's at 5 and then it goes down to here. OK? So, thus, there is a local maximum at those corresponding points, OK, corresponding uh X and Y values, OK. So we can say that our local maximum at X equals -2, -2, which is 6, OK. And At F of 5, OK, where X equals 5, which also equals 6. So these are our local minimums and maximums. Finally, let's try to find the absolute extreme values. Do we have any absolute extreme values here? Well, that means we're looking for the largest value of FFX and the smallest value of FX. Starting with the largest, notice that in our graph, OK, the largest value of FFX in the given graph is at X equals 11, OK? And where X equals 11. OK. Then why, if we come across here, we can tell that Y will be equal to 7.5. It's halfway between 7 and 8. So thus, that tells us that our absolute maximum. Yeah absolute maximum. I F 11 equals 7.5, OK. And now for our absolute minimum, this one is a bit interesting here because you notice that it would appear the smallest value in our graph is done where X equals 5, but it's not defined. So that means there is no absolute minimum value of the function, OK? So there's no absolute minimum. Thus, in summary, our function is increasing on the intervals -7 to -2, union with 2 to 5, union with 5 to 11. It's decreasing on the interval from -2 to 2. The local minimum is F of 2 equals -6, while the local maximum is F of -2 and FF 5 equals 6. The absolute maximum is F of 11 equals 7.5, and it has no absolute minimum. Thanks a lot for watching everyone. I hope this video helped.

Identifying Extrema

In Exercises 15–18:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

Key Concepts

Increasing and Decreasing Intervals

Local and Absolute Extrema

Critical Points

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