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The function f is differentiable ad increasing on the interval 0 _ I < and the graph of f has exactly two points of inflection on this interval Which of the foll...

Question

The function f is differentiable ad increasing on the interval 0 _ I < and the graph of f has exactly two points of inflection on this interval Which of the following could be the graph of f the derivative of fSubmit

The function f is differentiable ad increasing on the interval 0 _ I < and the graph of f has exactly two points of inflection on this interval Which of the following could be the graph of f the derivative of f Submit



Answers

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is shown.
(a) On what intervals is $f$ increasing or decreasing?
(b) At what values of $x$ does $f$ have a local maximum or minimum?
(c) On what intervals is $f$ concave upward or downward?
(d) State the $x$ -coordinate(s) of the point(s) of inflection.
(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

Suffer on the given breath. We know that Ah, if prime is positive on, uh, these some intervals and these negative on the other intervals. So f is increasing home 1 to 6 and union with a tree infinity, and that is decreasing. 00 to 1. Union weighs 6 to 8. That means we have a local mix at X equals 26 And there we have a local Milliman. So I have to Local minimums at X equals two wind exceed coast rate. Now for the inflection part, we know that if prime f prime misty crazy. So if my is increasing phone 0 to 2 on the Ruutel five and all seven to Infinity, that means it's come that every is concave up. Because if if prime is increasing, that means the second the purity of his positive this country. Bob and F primes Deke raising long 2 to 3 union with 5 to 7 sitcom kept on. And let me is we have some infection point. Um at X equals 2 to 35 and seven because at each point, contribute ease are changing. So can um, GREss America sketch dysfunctional coordinate. Firstly, Laboard's off the intervals for one. The 12 um three, 45678 Okay, so if wrong 0 to 1, it's conch, A iced tea crazy in conclave town. So it looks like this. So there's a creating a point. No cold medium. No, it's con cave down and the increasing. Now we have a inflection point at X equals two. That means that can carry it. Is that different? Um, there is increasing. Come, keep up that you use this eso at X equals three. There's another inflection point that means the continuities there continue t will be different. It's known from 3 to 4 is increasing and can keep up. And, uh, from photo foot, actually, from 3 to 5 from 3 to 5, it's increasing in a conclave up. So we can just direct the graph this No, it it exit costs of five years. And now the inflection points in the from 5 to 6. It's increasing the contest. Um, and at six, we have a quick appoint a local man, Timon No, from 6 to 7. He crazy and calm down. Now, if seven continued its infection point, so from seven to hate, it's decreasing in the conclave. Up now. Eight. Yes. Ah, it's a local. So from eight to infinity, we have, uh, can keep popping increasing. It looks like it's

Okay, so we're given I draw the graph of F crime fellow and were being asked about the different things that the function So for a were asked about what already interval and increasing or decreasing. So for this we simply look at whether the function is positive s O that occurs when the function is above the graph. Because remember, this is a crime. And this occurs from zero two two and from forty six. And then this is actually eight right here that this is a sorry about that. And this is from eight to infinity. So this is increasing and then it is decreasing everywhere else. So that would be from two, two, four and sixty eight. So it is decreasing here. Ah, for local max in man, we have local backs and men's. Okay, So local max occurs when it goes from positive to negative. So this occurs from process negatives of Mexico's too so local Mac two. And then it occurs again at night at six. Because it's going from positive to negative at six, two and six and then local men occurred when it goes from negative to positive. So this is from negative to positive. So that's four and negative to positive again at eight. So this is it for sea on one two intervals is a conclave. Absolute. So it is. Khan gave up when the ah slope s So when the functions increases with the equation So cardio occurred. Ah, from the sea when the slope is increasing. So this is you could see this is decreasing and then it's increasing from three to six. It is increasing. This can't give up and then it again increasing from six to infinity. And then it is Kana Cave down when the slope is negative, so decreasing This is from zero three and that is it. And then for d were asked Ah, there any inflection point? So there's an infection point of Mexico's three because the sign changes, It goes from a negative slope to a positive floats. And so that means the second derivative is changing and by definition, infection, pork ribs. When the Khan Cabinet changes or the sign of the second derivative changes, that's X equals three and then for e. We're going, Teo Graff said this is a prime for being asked Graff Ah, the graph of f and we'Ll do it on the next page and we're assuming that half of zero zero So we'LL start right here. And basically it increases and decrease is giving us a local men right here. And then it goes off to some Ah, remember, has a whole goes off so positive and it goes up to a very high point wherever this point, maybe s O goes after that point and make This is not exactly the skill it comes down. There's another local big the increasing. So it's not exactly the best graft, but this is a right here. These are like critical point. This is sick and this is a vertical ascent. Oh, I mean, not every glass it Oh, sorry. Just a I'm just drawing this to show you that there is a, uh it's sharp corner here is continuous because I'm so sorry. This is continuous because it's told us. But the graph of a continuous function f so this is still continue to just have the sharp corner that's quite a derivative doesn't clear. And then this point right here is for because again, this is a local backs and bitter this another local. This is a local better. This is a local back, though that we have identified him early admissions Go

Okay, so this is a graph of prime right here, and we're being asked about different things about the function F so it's just go ahead and jump right into it. So we're being asked where efforts increasing, decreasing. So we just simply look at where the function is above the exactly for positive. So that is increasing for that occurs from Ah one, two, six the one, the sixth and eight to infinity. And then it is decreasing where it is below the X axis. So that's from zero to one and then from sixty. Oh, that looks like an infinity, Uh and then for be where they asked the local maxim men So local back and local big So America our local backed occurs when it goes from positive to negative and then from right and the local medical from negative to positive. So this is going around negative to positive. So that's a local red at one and occurs at it again. And then a local Mexico's at six. Because going from positive to negative and that con cave up occurred when the slope is positive or increasing. So that's occurs from zero to two and three to five and for seven to infinity and then call it a cave down occurs when the slope is negative and this occurred between two and three and five and seven. The inflection point occurs when the slope changes and the slope changes only when there is a some sort of local back from manicuring so gently where the tenderloin is zero. So if you look at the point where potential injury you have two, three, five, and seven So those are inflection point practical two, three five, seven to draw thgraf Of f What we do is we had just drops of access Maxie's So we are told that it is decreasing from zero one and then we have a local men at one and that it is kind of gave up from zero to two. So it's gonna look something like a U. But going down, what's going on? You have a local band right here want this will be one and then I believe that it increases from one to six is increasing and then it room it. Ah, it is Khan cave down for between no, from zero to still can't give up and then it switches that too, So it would look something like good. We'Ll continue to rise and that will stop it. Three. Cliff they don't go up because ah, see between So we're still at the way we go. Two sixes increases over all the way to six. It was increasing all the way to six that we have local Max. It's six, remember? Ah, little back. That's six. So that's why I stopped right there and that it goes down from there From six to eight, Stuff of sixty eight is decreasing and there's a local mid at eight. So we're going to go all the way down and it's gonna and it comes down to be some kind of local bed and then I think it goes. Officer increases from eight to infinity. So this has to go all the way. I like it. This will be six one. This is a This is a rough sketch of half through all of the information given to us

So because we are not specifically given a graph and the website, we don't have access to the graph. Um Instead, what we can do though um is discussed how we can take the graph of the derivative and no uh important information about the original graph. So for example, let's say are derivative graph look something like this for example. And it's a very simple, there's a very simple example. Um but it does show some important at attributes um or something like this perhaps. And then um we'll zoom in. So that way it's easier to see. So with this we see that because this is the derivative graph, we know the function will be decreasing when the actual graph is negative. So the original function decreases when the derivative graph is negative and the original function is increasing when the derivative graph is positive. We know that um there's going to be local minimums when the graph is decreasing and then hits this point and then increases. So this would be the local minimum. And then conversely this should be a local maximum. Um And then if we know that f of zero equals zero, a possible graph that we could have would be execute my to act in this case. And that's using this knowledge because we know that if we had plus two for example, the F of zero would not be zero. But by giving this initial condition, we can take what's known as the anti derivative of the function. Um and see the general shape of the curve will be representative of the actual derivative graph that we've seen before. And this is going to be the general process by which we answer these problems. Um and take the derivative to find some important values. We see that um the derivative, if we talk about the derivative and the second derivative, if this is our original function, the derivative is going to be greater. If it's positive like this and the steeper the positive slope, the greater it is. However, the steeper the negative slope, the lesser it is. If our graph is concave up like this, the second derivative would be positive and if the graph is concave down like this then the second derivative would be negative. So those are all important information in determining the value of the function, The 2nd derivatives, first derivatives. Um So it's gonna be the information that we use a hand


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