5

Find the absolute maximum and minimum values of f on D. (a) f(x,y) = x2+y2 2x; D is the closed triangular region with vertices (2,0), (0,2), and (0,-2)_ Answer: max...

Question

Find the absolute maximum and minimum values of f on D. (a) f(x,y) = x2+y2 2x; D is the closed triangular region with vertices (2,0), (0,2), and (0,-2)_ Answer: max: f(0,+2) = 4; min: f(1,0) = -1 (b) f(x,Y) = 4x?+Y; D = {(X,y)lx? +y2 < 1} Answer: max: f(+1,0) 4; min: f(0,0) = 0 Find three positive numbers whose sum is 60 and whose product is maximum Hint: The problem can read: max *YZ , subject to (xY,zi Thus for example it can be reformulated a5; maxxy(60 ~y) KY with each component being pos

Find the absolute maximum and minimum values of f on D. (a) f(x,y) = x2+y2 2x; D is the closed triangular region with vertices (2,0), (0,2), and (0,-2)_ Answer: max: f(0,+2) = 4; min: f(1,0) = -1 (b) f(x,Y) = 4x?+Y; D = {(X,y)lx? +y2 < 1} Answer: max: f(+1,0) 4; min: f(0,0) = 0 Find three positive numbers whose sum is 60 and whose product is maximum Hint: The problem can read: max *YZ , subject to (xY,zi Thus for example it can be reformulated a5; maxxy(60 ~y) KY with each component being positive. From this; You may conclude x Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane 2x Sy 30. Clue: Try to use the hint given for Problem 14.81.



Answers

Box with vertex on a plane Find the volume of the largest closed
rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane $x / a+y / b+z / c=1$
where $a>0, b>0,$ and $c>0$ .

Okay, So, trying to maximize the volume of erecting given the X over April's wherever be for zero C equals one and a and being see must all be greater than zero. So f sensor china maximized volume was going to be equal to X y z and Delta F is gonna be equal to a lender times Delta G. So let's take the partial derivatives of each So first and I had direction we're going to get why Z I had Dan is gonna be place xz jihad plus X. Why, k that's gonna be equal to Landover A plus we'll and over be possible and oversee. So listen, make these on I j k directions, by the way and endless Eke said the components equal to each other. So why y Z is going to be able to land a xz is going to be good to land Toby X Y z and you go to learn to see Oversea Okay, so now I'm gonna saw Freelander. So what? We are a what would be the sea I z a is really good to Atlanta xz be what you could sell out toe ex lie sees no evil try and it So it clearly they're all equal to each other. But unless for instance takes said to it and he got each other. So why Z a is going to go to X Dizzy Be alright, Clearly we can cancel disease. And now this offer. Why? So why is gonna be egos to XB over a Now let's take two other ones that we could teach other and child get it in terms of X. So let's do, for instance, us do y z and X Y c So why is he a Because we're related to X Y c? Why is the cancel out z a is gonna be able to x c Z is equal to X C over a Okay, so now it's Fergus. Why dizzy back into the G equation g of X? Was the equation X over a plus? Why? Why would be so XB over a times would be clear. You see the bees cancel. All right, I will just go ahead with your day action for now and then. Plus, Z was gonna be X c over a time See on the bottle is going to be even someone so clearly you get excellent. A plus X over a plus X over A is equal to what, three x over a It's equal toe. So X is gonna be a over three. So since axity good today would really solve for y and Z, why is gonna be good to a over three times? Be over a is cancel up. Why is able to be over? Z is gonna be able to a over three times see over a the A's were going to cancel out and Z is gonna be equal to see over three. So the final answer you're gonna get a over three, be over drink, see over to

Mhm. We want to find the maximum volume of a rectangular box with three faces, the coordinate planes and a Vertex on the Plain X plus y plus e equals one with it's three coordinates. Positive. That is the point. He's on the first OC tent and on the plain X plus y plus Z equals one. So the confuse rations that we have three phases of the box lying on the coordinate planes X y y Z X Z and then the point in the first accent and on the plain X plus y plus equals once you find in the shape of the mm box. So we know that the volume, how the boxes X plus y plus e, which are the dimensions of the box an using the fact that the Point X Y Z is on the plain X plus y plus Z equals one. We get Z equals one minus x minus. Why? And so the volume is a function of two variables X y, and it's equal to X times Y times one minus X minus Why? Which is equal to X Y minus X square? Why minus eggs. White Square. So this is the function of two variables. Yeah, that we want Thio maximize X Y many six square. Why many six x y square. And we have also uh huh X, y and Z are positive. Okay, then we want to find Yeah, Okay, Uh, the absolute maximum. Yeah, off the function v of two variables given by this expression here, four eggs And why positive initially disease The statement of the problem. But we can restate it and noticing the following. Yeah, so notice that mhm X Y being positive, Okay. And, uh, x y z being on the plane X plus y plus equals Wang implies that well, eggs iss less than one and positive. And why that? Because thes some here is greater than X. He's also greater than why so x get to be Listen, what? And that is because X and y are positive. That's a key thing, because in that X, y and Z are positive. So did some. Here is always greater than each of the terms off participating on the song. So X gotta be between 01 and also does be and why? So we had to have also why between zero and one. In fact, Zygo get to be between 012 So this condition help us to restate the problem this way. So we can re stage. Okay? Uh huh. Problem, Uh, find the absolute maximum. Okay, okay. Off vfx y given by discretion above on the closed re tangle. Okay, okay. Given by these inequalities. And that is because we are including here. The province are radically original Want to solve. So we know that the solution of this problem is gonna be certainly the solution of this one. If the solution is not on the boundary, but we will be the cases we see later. So, um I mean, this is our problem to solve. We know in this case and that the important thing is that with this statement is a problem. We know that there is a solution that is, we know the function B has an absolute maximum on the's rectangle. And that's because the rectangle is close and bounded. NBC's continues, so we have a solution of thes restatement of the problem. And we also know that thes absolute maximum get to be attained at a critical point or on the boundary of the retake. So we start with the critical points. Yeah. Of being No. Yeah. And for that to calculate those critical points we get Thio, calculate the partial derivatives off we and we obtain that. Be partial derivative. Respect to X is equal to why minus two X Y minus White square and the partial derivative of fever respect. Why Sequel to X minus X square minus two x y Remember, we are differentiating this expression here, and that's a These are the result of that differentiation. And now we get to solve thes system of equations given by why minus two x y minus y squared equals zero and X minus X square minus two x y equals zero And here we can take come factors in the first equation Why? And we get white times one minus two eggs minus Why zero? And then we have x times taking out factor X in second equation X times one minus X Man is too I equals zero. So he has the four factors We have factories Asian off Liberate the first equation second equation so we can combine the factors. For example, X and y zero or X equals zero on the factor one minus two X minus Y equals zero The factor one minus X minus two X minus two y equals zero and Y zero. Or the two factors parenthesis equal to Syria. We have combinations to make here. But what's important is that we can simplify in those cases pulling away. So we can say that if X equals zero and why cause here, which is one possibility, or we have X equals zero and being X equals zero. Here we combined it. We have combined it with Why So we combine out now with this factor inside parenthesis here. So one minus two x minus Y equals zero. Yeah, or Thea. Other combination you want to include here is white zero that is now take in I zero and take Weichel zero. Then we combined with these other factor. So why 10 and one minus x? Yeah, minus toe. Why equals C room? What happened in all these cases? Well, if we look at the formula B, which we will right here to remember here a suicide note to have the suppression of V again here well be fixed y sequel to X Times y times one minus six minutes. Why? To have it at hand so you will see the expression of V. If we have. We are in the case. X and y equals zero, this one. Then it's clear that the zero, because we have X times one times something else of X times y become zero. So the function is no same thing when X and this factory zero because x zero. And here again, why zero? So the function is no. So we get be of X Y in either of these cases is equal to Syria. So he's becomes the value to be considered up. Yeah, to find the absolute maximum. So the three cases are simplified like that. Now we have the other case, which is, if you like the factors here in the system of equations, the two factors inside parenthesis in both equations. So there's 21 so we can say that. Okay, this combination is system one minus two x minus y equals zero and one minus X minus two y zero, which is the same as the system two x plus y equals one and eggs when plus two y puts one in that can be modified this way. We, uh okay, left the first equation asses and the second equation removed to play both sides by the negative to we get 92 times sex minus for why it was negative. Two. And it will soon up these two equations, we get negative three y equals anti one. So we get white, Which one third? Yeah. And from we can take one of any of the equations of the system we can say, for example, X this question here, X bless to wipe was one yet X equal one minus two eggs. That is one minus two times one third. That is one minus two thirds. And that is one third again. So eggs is one third. And it's important to note that his solution is valid because it's critical point, because it's on the interval 01 which is a condition for eggs on why this moment? So uh huh. Right and eggs and why all right in the interval on the interval? Uh, you know what? So they're valid Critical points and f Sorry, B if the function of the the volume at thes point, he's one third time's one third times one minus one third minutes. One third, That is one third time's one third times one minus two thirties Once thesis one third again. So we one third to this. Do you weigh Have won over 27. So the volume at this critical 0.1 3rd, one third is 1/27. So this is, uh that is that is all for the critical points. So we have found the critical points. Uh, seriously, you are this one here. One third, one third. And maybe there are others that because there are these two conditions, But but in this case, remember, we don't need necessarily to write down the critical points, but the value at those groups critical points in that is why this case here group all these possible critical points where the volume is equal to zero. So we have thes to value zero and this one here to be considered at the end. So now we go to the boundary of T retain. Okay. We get to Sam in the function there, okay? Yeah. And we are going to do that. We're going to do that. Um, yeah. All right. Going for each edges for each h of the rectangle. But we can again groups some edges together and we start with edges. X equals zero. And why? Because serum one and the edge Uh huh y equals zero. Yeah, and x between zero and one. In those cases, we have p of wine. The first case. Because in this first case, why is the only variable we get Because exit replaced by zero, it gets hero or P of X. Consider, here is a function of X. We go through your place, Weichel zero. So we get zero. In both cases on DSO, we have to consider Deve earlier zero, which we already consider of law. Now we go, we go to the edges. Oh, Uh huh X equal one. And why between serum one and y equals one on eggs between 01 Yeah, and in those cases, we have be a wind. The first case because extra replace what you want. And if we see the formula of me with the volume here and we replace X Y one So we have consolation here of thes two terms and we get this is once we get white times native wise, so you get negative White Square, Andi symmetrically we get V of X equal negative square when y equals one. So in either case, we will have to consider two values negative one and zero. Because for these two functions, the maximum and minimum is are attained, are retained at negative at one and zero and values are negative one and zero. And there, actually Mom and I mean a moon values you are Maxim. First we get zero and in a month 2nd, 31. So these are two valves to consider. Zero is all very consider. And now we at the value negative one. And that's all the ages that we have. So we can now collect all the values we have to consider. So we can say that formal, the values we have to consider they are negative. 10 is to hear 1/27 and see room. Yeah, So there are only these three values and we can conclude way can see easily that this is the absolute maximum. So we conclude, Yeah, that the absolute maximum off the following function V of X y ese 1/27 and that value is attained at the critical point. It's very important to notice that is not attended The battery of the we tangle, but at a critical point one third, one third Mhm as we saw above. So we get now the value of Z from equation of the plane where point is X plus y plus C equals swan. Get Z equals one minus X minus. What? That is one minus one third, minus one third. That is one when it's two thirds and that is one third. So Z equals one third also, and we can now translate the result. In terms of the statement of the problems, we can see that the maximum volume mhm off a rectangular box with three faces in the coordinate planes. Okay? Yeah, right. And verdicts in the first opt int. Okay, okay. That in on the plane X plus y z equals swamp the that actual vaccine volume is 1/27. Okay. And the box is is just a cube. Okay. Thank you, Bob. Offsides One third. And we know it's a cube. Simply because we have three dimensions of the box are equal. So we have a cube and the volume is 1/26

Hey, it's Claire's when you read Here. Here, we given the equation three plus X lie minus X minus to lie. They're in Fort Houses that there is both an absolute maximum and minimum. So we're gonna find the critical points off. So we get X is equal to my lettuce. One why is equal to X minus two circuit of employees for this is to call no one and I took on the line is equal to one. And in step two, we're gonna look at the values of F on the boundary of D. And this consists of three line segments. Line one access equal to one that's in the domain. So for willing to Why is this so since the domain of one bye for X line three why minus fail is equal to or a minus zero over one minus by time ticks. So why is equal to negative X plus five? This isn't the domains of war. We're all you know. Why so online? One we have X is equal to So this is a decreasing function. So it's gonna be a minimum at were lying. One. This is a minimum. Why equals four in a maximum at, like was served. When calm you zero is equal to Yeah. When you call my floors, you get to negative too. So we're lying to We have, uh all right. It's common with zero is equal to three minus x No domain one. Why? For Epps, this is a decrease in function. So it's a minimum. X equals five and a maximum X equals one. So when the zeros equal to two in a while, sir. Rasi to negative too. Then for line three. Here we have. Why is he too negative? Negative X plus five. So we're plugging ex Common Negative X plus y. I'll get negative Explorer. They're gonna scaring. Really six x plus seven. This isn't the domain, uh, zero for the value. Why, then the function as a critical point X equals three and its point of maximum by the second derivative test. So, uh, recall Mater's equal to Harris is we don't have much space One call more floors able to negative too, in a five common zeros equal to editor. So we know that three comma too and one comment zero are the absolute maximum and the points one comma, four and five comma zero on the maximum values of the domain is too. An absolute minimum on the Domaine de is

So giving that we have F. So giving F X. Y. The function F. To be equal to three plus xy My next eggs -2 Y. Then you have the sets the sea vehicles closed triangular ranging with with veterans is You have 10 50 then 14. So what we want to do is find the absolute minimum and absolute maximum values of the given function. Only this giving six d. So what we first do is we face capital. It's so we talked. Leaves the critic our points the critical points. Boys basic appoints. It's F. of X. is equal to zero. And yeah F. X equal to zero and F. Y. Also equal to. So so what we do then is find the partial derivative with respect to X. For this. So if with respect to X will be equal to. So we have why my next one then have respect to why will be equal to it's minus soon. So then at this point we are saying this should be equal to zoo. So they this implies that you have your F. Of X equal to zero. That is why my next one equal to zero. So then this implies that's why it's equal to one. Then X -2 equals zero. Which implies that's our eggs equal to two. Now not only we need to look at this critical point, we need to look at the boundaries of the giving safety as well. So the equation of the line theory. So now we look at the equation equation of the line the line to. So giving do we give in points? Let's see 1 4 and 50 is going to be you have Why to be equal to 0? My next four Divided by five. Mine it's one Yeah it's my next one last four. So away here. So why did so why why will be equal to negative X Plus one Plus 4? Because this will give us -1 here. Sometimes there's negative eggs. This will give us positive and this will be equal to negative X plus five. So this implies that if so f at the point X end negative X plus five will be equal to to me lads, eggs you have youtube X last five- X -2. We have negative X plus five. So this vehicle to three minus X squared last five eights. Mine it's eight. My plus two eggs. Then my name is saying. So you have this and that's a common. So you have this is going to be negative X squared five minus this. You have four plus 26 X plus six X. My nets. So so then the pasha divisive of this with respect to X would be cool to. So f with respect to X. Here we give us These vehicles -2 X. Last six. So at a critical point yes saying that this should be equal to zero. So they this implies that You have negative two eggs to be called to -6. So then this implies death X. It's equal to three. Which implies that then why will be equal to soon? Soon. Therefore now we have more critic our points. So so okay Jessica. Oh critica poise. Ah You have 21. You have 10 50. You have 1 4 then three soon. So then this implies death. If see how F. At this point to one Will be equal to one. Then f. at the .10 Will be equal to two. Then F. Are the points five zero. It's equal to -2. Then F. at the .14 it's equal to negative to us. Well Then f at the .32 It's equal to two. So then you realize that the absolute so the absolutes months and more maximum is going to be this and that. So you have F 10 to be equal to F. 3 2. Which is equal to soon. Then our absolutes absolutes minimum minimum it's -2. So you have f of 50 F. At the 500.50. Then F. At the 0.14 which is equal to negative sue. As a finer results


Similar Solved Questions

5 answers
(d)dx V9-x(x = 3sin(8))
(d) dx V9-x (x = 3sin(8))...
5 answers
Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem y' =7x2 +2y2; Y(O) = 1The Taylor approximation to three nonzero terms is y(x)
Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem y' =7x2 +2y2; Y(O) = 1 The Taylor approximation to three nonzero terms is y(x)...
5 answers
Bpoint)A 4 MC point charge is placed at the origin as shown in the figure above, where a = 1.2 cm b = 2.8 cm; c = 1.7cm, and d = 3.3cm Find the potential at the points A, B and C: VA VBVc
B point) A 4 MC point charge is placed at the origin as shown in the figure above, where a = 1.2 cm b = 2.8 cm; c = 1.7cm, and d = 3.3cm Find the potential at the points A, B and C: VA VB Vc...
5 answers
Sec 16.5 Rotacion y Divergencia= Problem 3PreviousProblem ListNextpoint) Let (4a'€ + 4ay?)i (423 2ay)j (22 + 4.2 + 4y2)k.(a) Find the value(s) of a making div F _ 0(Enter your value, or if you have more than one, enter a comma-separated list of your values:(b) Find the value(s) of a making div F a minimum(Enter your value, or if you have more than one, enter a comma-separated list of your values )
Sec 16.5 Rotacion y Divergencia= Problem 3 Previous Problem List Next point) Let (4a'€ + 4ay?)i (423 2ay)j (22 + 4.2 + 4y2)k. (a) Find the value(s) of a making div F _ 0 (Enter your value, or if you have more than one, enter a comma-separated list of your values: (b) Find the value(s) of ...
5 answers
Give examples of sequences one in each case, having the following properties (you don need to prove anything here): marks] (an) is convergent; but not monotone; marks] (an) and (a5 are both unbounded; marks] (4u) is not convergent, but it has convergent subsequences, (bu) , (cn) land K(du) such that: bn 0 and d, [1 marks] (an) is not convergent, but (2a% Sau) is convergent_
Give examples of sequences one in each case, having the following properties (you don need to prove anything here): marks] (an) is convergent; but not monotone; marks] (an) and (a5 are both unbounded; marks] (4u) is not convergent, but it has convergent subsequences, (bu) , (cn) land K(du) such tha...
5 answers
Question 7political scientist surveys 32 of the current 32 representatives in state $ congress_What is the size of the sample;What the size of the population:Next Question
Question 7 political scientist surveys 32 of the current 32 representatives in state $ congress_ What is the size of the sample; What the size of the population: Next Question...
5 answers
The molar solubility of CuSCNis 4.21 1.77 * 10-13 2.98 10-19 6.9 < 10-4 842 * 10" 3,4 * 10-13Man PUrc - Water, Calculate the Ksp = for CuSC):
The molar solubility of CuSCNis 4.21 1.77 * 10-13 2.98 10-19 6.9 < 10-4 842 * 10" 3,4 * 10-13 Man PUrc - Water, Calculate the Ksp = for CuSC):...
1 answers
Use equation (2.6) to help solve the IVP. $$ y^{\prime}=2 y(5-y), y(0)=4 $$
Use equation (2.6) to help solve the IVP. $$ y^{\prime}=2 y(5-y), y(0)=4 $$...
5 answers
Determine whether the given statement is true false;Iis given that the row reduced echelon form ofthe matrix Athe matrix R=1 3 0 0(e system of linear equations 3+0+52+w=6 an 2+W=2 20*8+2+4w=13 082-2w= Kpes Iniinitely many solutions:Aaeaone: Tne False
Determine whether the given statement is true false; Iis given that the row reduced echelon form ofthe matrix A the matrix R= 1 3 0 0 (e system of linear equations 3+0+52+w=6 an 2+W=2 20*8+2+4w=13 082-2w= Kpes Iniinitely many solutions: Aaeaone: Tne False...
5 answers
Arock & dropped from Ihe top of 4 256-foot cliff The height leet of Ihe rack polynomial 256 - 16r' Faclot this expression above Ihe water after [ seconds modeled by the comptetely256 - A16 -[
Arock & dropped from Ihe top of 4 256-foot cliff The height leet of Ihe rack polynomial 256 - 16r' Faclot this expression above Ihe water after [ seconds modeled by the comptetely 256 - A16 -[...
1 answers
In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes the given point. $$f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)$$
In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes the given point. $$f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)$$...
5 answers
Question 33 ptsOne end of a polypeptide chain is shown below:HzNOHIs this the N-terminus or the C-terminus of the polypeptide? Select |What is the next backbone atom in the chain (i.e what kind of atom would be revealed if the squiggly line were moved over by one position)? (Select ]
Question 3 3 pts One end of a polypeptide chain is shown below: HzN OH Is this the N-terminus or the C-terminus of the polypeptide? Select | What is the next backbone atom in the chain (i.e what kind of atom would be revealed if the squiggly line were moved over by one position)? (Select ]...
5 answers
Independent random sampling from two normally distributed populations gives the results below Find 9g% confidence interval estimate of the differerce between means of the two populations_117n2 -X2 = 10802The confidence interva (01 - H2) (Round t0 four decimal places as neeced
Independent random sampling from two normally distributed populations gives the results below Find 9g% confidence interval estimate of the differerce between means of the two populations_ 117 n2 - X2 = 108 02 The confidence interva (01 - H2) (Round t0 four decimal places as neeced...
5 answers
Izou Eeeinn2 pointsThe DeWitt Company has found that the rale of change of its average cost of producing 7160 a product is € Iwhere x is the number of units and cost is in dollars. The average cost of producing 20 units is S33.00 Find the average cost of producing 160 units of the product Round t0 the nearest dollar.0 S98 S91 0 S93 S96S103
izou Eeeinn 2 points The DeWitt Company has found that the rale of change of its average cost of producing 7160 a product is € Iwhere x is the number of units and cost is in dollars. The average cost of producing 20 units is S33.00 Find the average cost of producing 160 units of the product R...
5 answers
{cutia uscTu P astics.Ihelmst S COcro deancIndustria preparation There Gte [0 Sucd= acerylene and cm hydrozide: Latet rejcl (Dacnlic 2CICImmedute precursorCC,(s)+2 H;Ol9) -C,H,(9)+Ca(OH),(s) acety Cne carbon dioxide and Water read Fcont SEo{ngm acrc Jd:6C,H,(9)+3CO_(9)+4 HzOl9)-SCH,CHCO_H(9) chemica Fqubucn the Drodunctlor aoTIc jCio Wrile theBe sure Your cquation balanced and carbon dioxidcTalclum camdioc;0-C
{cutia uscTu P astics. Ihelmst S CO cro deanc Industria preparation There Gte [0 Sucd= acerylene and cm hydrozide: Latet rejcl (D acnlic 2CIC Immedute precursor CC,(s)+2 H;Ol9) -C,H,(9)+Ca(OH),(s) acety Cne carbon dioxide and Water read Fcont SEo {ngm acrc Jd: 6C,H,(9)+3CO_(9)+4 HzOl9)-SCH,CHCO_H(9)...

-- 0.020725--