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The graph of the derivative $ f' $ of a continuous function $ f $ is shown.

(a) On what intervals is $ f $ increasing? Decreasing?

(b) At what values of $ x $ does $ f $ have a local maximum? Local minimum?

(c) On what intervals is $ f $ concave upward? Concave downward?

(d) State the $ x $-coordinate(s) of the point(s) of inflection.

(e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

a) $f(x)$ is increasing for $x \in(1,6) \cup(8, \infty)$

$f(x)$ is decreasing for $x \in(0,1) \cup(6,8)$

b) $f$ has a local maximum at 6 and local minima at 1 and $8 .$

c) $f$ is $\mathrm{CU}$ on $(0,2) \cup(3,5) \cup(7, \infty)$ and $\mathrm{CD}$ on $(2,3) \cup(5,7)$

d) $f$ has inflection points at $x=2,3,5$ and 7

e) see the explanation for solution

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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