5

Let the domain of $f$ be an open interval $I$, and assume that $f^{-1}$ exists. Suppose $f^{prime prime}$ exists on $I$ and $f^{prime}(x) eq 0$ and $f^{prime prime}...

Question

Let the domain of $f$ be an open interval $I$, and assume that $f^{-1}$ exists. Suppose $f^{prime prime}$ exists on $I$ and $f^{prime}(x) eq 0$ and $f^{prime prime}(x) eq 0$ for all $x$ in $I$a. Suppose $f$ is increasing on $I$. Show that the graph of $f^{-1}$ is concave upward on its domain if the graph of $f$ is concave downward on $I$, and the graph of $f^{-1}$ is concave downward on its domain if the graph of $f$ is concave upward on $I .$ (Hint: Use Exercise 73.)b. Suppose $f$ is decreasing

Let the domain of $f$ be an open interval $I$, and assume that $f^{-1}$ exists. Suppose $f^{prime prime}$ exists on $I$ and $f^{prime}(x) eq 0$ and $f^{prime prime}(x) eq 0$ for all $x$ in $I$ a. Suppose $f$ is increasing on $I$. Show that the graph of $f^{-1}$ is concave upward on its domain if the graph of $f$ is concave downward on $I$, and the graph of $f^{-1}$ is concave downward on its domain if the graph of $f$ is concave upward on $I .$ (Hint: Use Exercise 73.) b. Suppose $f$ is decreasing on $I$. Show that the graph of $f^{-1}$ is concave upward on its domain if the graph of $f$ is concave upward on $I$, and the graph of $f^{-1}$ is concave downward on its domain if the graph of $f$ is concave downward on $I$. c. Suppose that $a$ is in $I$ and that the graph of $f$ has an inflection point at $(a, f(a))$. What can you say about an inflection point for the graph of $f^{-1}$ ?



Answers

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.

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

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

From the given graph off the purity if off if we know the following information So party so increasing uh, interval will be wrong. 0 to 2 union with 4 to 6 and the union wave Hey, to Not because over on these intervals f prime Miss politics and for decreasing part It's from 2 to 4. Um and the union was 62 So from part A, we know there is a local. Yes, I'm local maximum look monuments. Local mexes at X equals 22 and X equals the six. No commitment is that X equals two for index equals Great, huh? No, for the posse, we are looking for the inflection concurrently, An inflection point. So, uh, from the graph, we know if prime used if prime mystique raising when 0 to 3 and there is increasing on 3 to 3 to six union always 6 to 9. So we need to exclude, um X equals a six year because And this points Not if it's not differential. Knowing eso if I've prime is decreasing. That means they have top a couple prime years this land zero So it's conclave conclave down and ah, if if prime is increasing so it can keep up. So we have lasting go where for one inflection points at X equals the three. So with this information, we can give a sketch for this function. So first relay boat off that intervals from 23 four, 26 and eight. Okay, so on the first interval, it's increasing and it's conclave down suits. You would be something like this and it's Deke Raising the conclave down and there's three is the inflection points, so you will change the concave. Biti's um, maybe you should put something like this. So at this point, X equal to three other communities will be different. Maybe something like this and the one X equals 24 its analogue greater point. So it has a horizontal tension line, the same SX e coast to So it's that the function starting crazy at this point. Okay, so we can see there's a trump at X equals 26 Damien's. There's a cast here. It's a special point. It's not differential anymore. Um, so you would be something like this. And again when X equals to eight. This Ah, quit your point. So it has a hard time. Potentially. So this is ah schedule


Similar Solved Questions

5 answers
083 pez Previous AnswersParametrize the portion of the cyllnder y2 + z2 16 that lies in the first octant inside the cylinder x?(Your instructors preter angle bracket notationfor vectors:){(u,v) =V 16 +12-16 +42 . V16wlthand 0
083 pez Previous Answers Parametrize the portion of the cyllnder y2 + z2 16 that lies in the first octant inside the cylinder x? (Your instructors preter angle bracket notation for vectors:) {(u,v) = V 16 +12 -16 +42 . V16 wlth and 0...
5 answers
HzC-(CH2)1o _OHH3C =H;c-(CHzho _ C_0-CH3HaC-(CHzho _CH3H3CHzC_OHH3C-(CHzho -OCHzCH3HacOH
HzC-(CH2)1o _ OH H3C = H;c-(CHzho _ C_0- CH3 HaC-(CHzho _ CH3 H3CHzC_OH H3C-(CHzho - OCHzCH3 Hac OH...
5 answers
#ertion $Poln}Find the aren ofthe region Inside the circle6sin 0 and outside the cardioid5in 01. Sketch and shade the region;AluciMotlComputatUrotte Conlent Cclattion
#ertion $ Poln} Find the aren ofthe region Inside the circle 6sin 0 and outside the cardioid 5in 01. Sketch and shade the region; Aluci Motl Computat Urotte Conlent Cclattion...
5 answers
14. Provide the structure 0/ the major organic pro} ucI of tl lollowing reaction Show detailetl mechanisn . Section: 10-3 CHSO0UHs15. Provide the structure of the major orgunic prod Uct of the following reaction. Show detailetl mneehanism , Section: 10-4Give the major product for the following rede won Shou detwilel mec hunistn for ech step. Section; 16-3Cu- CCH,TA:c_OH
14. Provide the structure 0/ the major organic pro} ucI of tl lollowing reaction Show detailetl mechanisn . Section: 10-3 CHSO0 UHs 15. Provide the structure of the major orgunic prod Uct of the following reaction. Show detailetl mneehanism , Section: 10-4 Give the major product for the following re...
5 answers
854 U w € [o, L) Oc4 dafv< Tw L3 7 C,17 6 Y Txp k ? E Fe C [o, L> (T) Ptat +hd Toounj Kmtmep (ii) IJ 7o7L IJ (Tn 7) F= ( - ^) F (iii) F,w d V ( Tw (v) Oore 7(4b) T(64} Gb € C Ce L]SifeFmn4Ab € ACc) 4n0 p €c [6s] (p ( Aib) ) 4P (rca) ,*(b))PlcaseSolvetul> Rrpla 6A4 Aal 8step bysteq clzalundel $+444 ce4 Duy ut ~d be (4mi2 Soma JineAUoCawfunderstendSome S$4PfAlsopleasc544 Clal pic_ Hur{MLiTMIman k,
854 U w € [o, L) Oc4 dafv< Tw L3 7 C,17 6 Y Txp k ? E Fe C [o, L> (T) Ptat +hd Toounj Kmtmep (ii) IJ 7o7L IJ (Tn 7) F= ( - ^) F (iii) F,w d V ( Tw (v) Oore 7(4b) T(64} Gb € C Ce L] Sife Fmn4 Ab € ACc) 4n0 p €c [6s] (p ( Aib) ) 4P (rca) ,*(b)) Plcase Solve tul> Rrpla ...
5 answers
Decide whether the figure is convex or not convexChoose the correct answer belownot convexconvex
Decide whether the figure is convex or not convex Choose the correct answer below not convex convex...
5 answers
What is the equilibrium pressure of Oz(g) over M(s) at 298 K?Po:atn
What is the equilibrium pressure of Oz(g) over M(s) at 298 K? Po: atn...
5 answers
Erom Chapter_LZ: Curentand_Resistance 16. According t0 Georg Simon Ohm how much current will be allowed to flow through a light bulb which has resistance of 60 Q pushed with an electromotive force of 12 volts? (Did (ohms) . if it'$ being you do your ohm work?) 5 amperes. 1/5 amperes. Zero amperes (not enough voltage
Erom Chapter_LZ: Curentand_Resistance 16. According t0 Georg Simon Ohm how much current will be allowed to flow through a light bulb which has resistance of 60 Q pushed with an electromotive force of 12 volts? (Did (ohms) . if it'$ being you do your ohm work?) 5 amperes. 1/5 amperes. Zero amper...
5 answers
Find the derivative of: $quad mathrm{G}(mathrm{x})=sin sqrt{operatorname{In}} mathrm{x}$, for $mathrm{x}>1$
Find the derivative of: $quad mathrm{G}(mathrm{x})=sin sqrt{operatorname{In}} mathrm{x}$, for $mathrm{x}>1$...
5 answers
An electric field is shown below: Assume the field is all in the plane of the paper: Where is the E-field the strongest?(i) and (iii)
An electric field is shown below: Assume the field is all in the plane of the paper: Where is the E-field the strongest? (i) and (iii)...
5 answers
0391J 4pple , Jently hung Irom Speing tht Stretche s bcm The Force CenStant SeringA)44Nfn BBNk-m 0.33 Nlo p)To N/n 044 Nln
0391J 4pple , Jently hung Irom Speing tht Stretche s bcm The Force CenStant Sering A)44Nfn BBNk-m 0.33 Nlo p)To N/n 044 Nln...
5 answers
What is the leading hypothesis for the Moon's formation? What evidence supports this hypothesis?
What is the leading hypothesis for the Moon's formation? What evidence supports this hypothesis?...
5 answers
11Iha lollowing mechonism has lxcn suggestexd lor Uho faxction 16O,' IVO 2h" Ioi OH" Slow Ol I,0 HOI Hio Fas Identily Ile rale law that Is consistenl with Ihis mechanism Points)MMOH IM"IRote # HH O,liimtnt'Rote VhoIIOHTRate MO.lli ]Rate = Khomtm7
11Iha lollowing mechonism has lxcn suggestexd lor Uho faxction 16O,' IVO 2h" Ioi OH" Slow Ol I,0 HOI Hio Fas Identily Ile rale law that Is consistenl with Ihis mechanism Points) M MOH IM"I Rote # HH O,liimtnt' Rote VhoIIOHT Rate MO.lli ] Rate = Khomtm7...
5 answers
Use Green's Theorem to evaluate the following line integral Assume the curve is oriented counterclockwise A sketch helpful. flzy-34x2 8) dr; where € is the boundary of the rectangle with vertices (0,0), (5,0), (5,3), and (0,3)Slzy-3,4x+8) dr = (Type an exact answer )
Use Green's Theorem to evaluate the following line integral Assume the curve is oriented counterclockwise A sketch helpful. flzy-34x2 8) dr; where € is the boundary of the rectangle with vertices (0,0), (5,0), (5,3), and (0,3) Slzy-3,4x+8) dr = (Type an exact answer )...
5 answers
Let x be a discrete random variable that possesses a binomialdistribution. Find the probability of exactly five successes in 7independent trials with the probability of success equals 0.45
Let x be a discrete random variable that possesses a binomial distribution. Find the probability of exactly five successes in 7 independent trials with the probability of success equals 0.45...
5 answers
Electrophilic 542 f = Nucleophilic subs addition Nucleophilic subst carbonyl(acyl Xfer) Carbonyl Conjugate (nucleophilic) Flimination nucleophilic addn addn Sy ! Nucleophilic suostIdentify the mechanism by which cach ofthe reactions above proceeds from among the mechanisms listed_ Use the letters for your anSWcTs WsiledSubmit AnswrorRotry Entire Groupmoro group attempts remainingPreviousNexi
Electrophilic 542 f = Nucleophilic subs addition Nucleophilic subst carbonyl(acyl Xfer) Carbonyl Conjugate (nucleophilic) Flimination nucleophilic addn addn Sy ! Nucleophilic suost Identify the mechanism by which cach ofthe reactions above proceeds from among the mechanisms listed_ Use the letters f...
5 answers
A can do a piece of work in 4 days and B can do it in half thetime. How longwill it take them to do the work together?Answer: 1(1/3) days. Please show all steps and work.
A can do a piece of work in 4 days and B can do it in half the time. How long will it take them to do the work together? Answer: 1(1/3) days. Please show all steps and work....
5 answers
In the case of measuring the volume of a liquid; you can measure it directly using graduated cylinder:To measure the mass of a solid,you need to determine the volume indirectly by what technique?
In the case of measuring the volume of a liquid; you can measure it directly using graduated cylinder: To measure the mass of a solid,you need to determine the volume indirectly by what technique?...

-- 0.019655--