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 minima and maxima, with labeled axes and a curve illustrating increasing and decreasing intervals.

Accepted Answer

Welcome back, everyone. In this problem, we want to analyze the given graph of the function F of X and the interval open bracket -7, 6, close the bracket to find the intervals where the function is increasing or decreasing and identify its local and absolute extreme values. And here we have our graph of F of X. Now let's break this problem down step by step. First, let's find where our function is increasing. By looking at our graph, we can tell where our Y values will be increasing, OK. Notice that our Y values are increasing here. OK, on our graph until it comes to a, a, a, a maximum point. Let me just highlight that again in red. It's increasing here again in red, and it's increasing where, again, it's highlighted in red. To know We can say that FF X is increasing on those three intervals. What will those intervals be? We'll notice here our first interval goes from -7 to halfway between -5 and -4. So that's -4.5. Next, it's increasing from -2 to 2. And then it's increasing from 4.5 to 6. So these are the intervals on which our graph is increasing. How about the next part of our problem? Where is FFX decreasing? Well, again, that will be the, the, the section where the Y values are getting smaller or low or, or less, OK? And now in this case, notice that they're getting less going from this interval again, again I've highlighted it in red. And on this interval, OK? And for both of those intervals, they go, the first one goes from -4.5 to -2, OK? So it's decreasing on the interval from -4.5 to -2. Union with the interval from 2. To 4.5, OK. So these are the points on our graph on which it is increasing and decreasing. Next, let's try to identify our graphs local extreme values. Now, that means the local sections are the points on our graph at which our graph is the lowest or highest, OK? So let's start with our local minimum. Now, at our local minimum, our graph is going to go from decreasing to increasing. Where is that happening? Notice here that it's happening at. Or at the point where X equals -2, and thus the corresponding value would be -12. So this tells us that F of -2 equals -12 is a local minimum. OK. Now is that happening anywhere else? Well, yes, notice here again that our graph goes from decreasing to increasing where X equals 4.5 and the corresponding Y value is 1.75. So F of 4.5. Equals 1.75 is also a local minimum. Now, what about our local maximum, OK? Well, again, at this point the opposite is happening. Our graph is going to go from increasing to decreasing. Where is that happening on our graph? Well, notice here, OK? That let me just uh make some space here. Notice here that we can tell that that is happening. At this point, OK. And from increasing to decreasing. At the point, it changes at the point where X equals 4. 4.5, and the corresponding Y value would be 0.5. So F of -4.5 equals 0.5 is one local maximum for function. And again, if we take a good look at our graph, notice that. When X equals 2, OK, it goes from increasing to decreasing. So that's another local maximum. So F of 2 equals 4. And we know it's 4 because 4 is where our function is defined, OK? And our function would be open where, well, notice here where our curve starts, OK? So that, that, that point is not included. Now, finally, let's try to find the absolute extreme values. And this means we're looking for the highest and lowest points on our graph. Now, what do you notice here? Well, notice that when it comes to our absolute minimum, OK. The lowest y value on our graph is -12. And at that point here you can notice that it's defined where X equals -7. So F of -7 equals -12 is a is the absolute minimum of our graph. What about our absolute maximum? Well, Again, if we take a good look at our graph, the largest x value, the largest value of F of X in the given graph, which would be at the top of our of our graph as an open circle. It's not defined. Thus, there is no absolute maximum value of the function. It's no absolute maximum value. Thus, in summary, our function is increasing on the interval from -7 to -4.5, from 2 to 2, and from 4.5 to 6. It's decreasing on the interval from -4.5 to -2, and from 2 to 4.5. The low minimums are F of -2 equals -12 and F of 4.5 equals 1.75. The local maximums are F-4.5 equals 0.5 and F2 equals 4. The absolute minimum is F of -7 equals -12, and there is no absolute maximum value. 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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