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Use the given graph $ f $ to find the following.

(a) The open intervals on which $ f $ is increasing.

(b) The open intervals on which $ f $ is decreasing.

(c) The open intervals on which $ f $ is concave upward.

(d) The open intervals on which $ f $ is concave downward.

(e) The coordinates of the points of inflection.

a. (0,1)$\cup(3,5) \cup(5,7)$

b. (1,3)

c. (2,4)$\cup(5,7)$

d. $(0,2)(4,5)$

e. (2,2)(4,3)(5,4)

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

So I've drawn a rough sketch of the graph that were given in this problem. And in part a we're going to try and find the intervals on which our graph is increasing. So if we look at the graph that I have here, you can see that from this point at 00 to this point at 1:03, we have a positive slope. So our graph or our function is increasing. So the first part of our interval is going to be from 0-1. And the reason that I know that uh we stop increasing at this point One comma three is because it's a maximum value, which means that the slope at that point or the derivative of our function at that point is going to be equal to zero. And so we're not increasing anymore. And after this point, after this maximum, we actually start decreasing until again this minimum where we have a slope of zero and then we start increasing again. And it might be kind of hard to see on my graph that I drew, but on the original graph it should be easier to see That at this .5:04 we actually have a slope of zero. Again, we have a critical point at this 00.0.5 comma four. So we need to realize that we're increasing from three and then at five we have a slope of zero, so we're no longer increasing here. So we're going from 3 to 5 for our second interval. And then if we look at our graph past this 0.5 comma four were increasing again until the end of our graph at this 40.7 comma six. So again, we're increasing from 5 to 7. And the reason that we don't want to include this point at five is because we have a slope of zero at five. And so our graph isn't actually increasing at that point enough for part B, what we want to find are the intervals in which we're decreasing. So if we look at our graph we can see that again. We said that we were increasing from 0 to 1 from 3 to 5 and from 5 to 7. So the only point on our graph are the only part of our graph that can be decreasing is in this interval from 1 to 3. And if we look at our graph we are decreasing. We have a negative slope from 1 to 3 with slips of zero at one and three. So we do not include those. So we have from 1 to 3 our graph is decreasing. And now for part C, we want to figure out what intervals are graph is actually concave upwards. So for this we're gonna want to look at inflection point and try to figure out where those points are. I tried to um highlight those points by putting them in red here and here and here. Um probably a little bit easier to see on the graph, but at these points we're going from being concave up or down to then being the opposite. So if we were concave up before this point, we're going to be concave down after, if we were concave down before it will be concave up after. So if we look at our graph, we can see that we're concave down before this first inflection point And at this point is when our graph changes from concave down to concave up. So we are concave up from one or sorry, from two To this point here at 4:03, so and after four comma three, since this is an inflection point, we're then going to be concave down for a little while. So the first interval in which we're concave up Is from 2- four. So you can say Your con gave up on the interval from 2 to 4 and then we look past four were concave down from four to this point at five. And then after this point at five, this is another inflection point. We're going to be concave up until the end of our graph at this 0.76 So we're concave up from this point here at two comma 2 to 4 comma three and then from five comma 4 to 7 comma six. So from 2-3 to and from five 27 And for part do you what we're going to try and find is where our graph is con Cape down. So That's going to be the other intervals of our graph or the interval from 0-2 and the interval from 4-5. And we can see this as we are concave down here until this inflection point at two or concave up. And then we start and then we become concave down again after this inflection point at four. And then our last inflection point here makes us con cave up again. So we're only concave down and this interval and from this point to this point. So our intervals are from 0 to 2 and from 4 to 5. And the last thing that we want to do is identify where these inflection points are. And I've already identified these since we had to to figure out where our graph was concave up and concave down. And so these are at this point here, This point here in this point here. So the points are 2,2, four comma three And 5:04. So to to 43 in five comma for