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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) $f$ is increasing where $f^{\prime}$ is positive, that is, on $(0,2),(4,6),$ and $(8, \infty) ;$ and decreasing where $f^{\prime}$ is negative, that is, on

(2,4) and (6,8)

(b) $f$ has local maxima where $f^{\prime}$ changes from positive to negative, at $x=2$ and at $x=6,$ and local minima where $f^{\prime}$ changes

from negative to positive, at $x=4$ and at $x=8$

(c) $f$ is concave upward (CU) where $f^{\prime}$ is increasing, that is, on (3,6) and $(6, \infty),$ and concave downward (CD) where $f^{\prime}$ is

decreasing, that is, on (0,3)

(d) There is a point of inflection where $f$ changes from

being $\mathrm{CD}$ to being $\mathrm{CU}$, that is, at $x=3$

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Missouri State University

Oregon State University

University of Michigan - Ann Arbor

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