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Let fkx,y) =xly+ ylnx Find f(1,1) , f(1,e), fle,1), and fle,e)...

Question

Let fkx,y) =xly+ ylnx Find f(1,1) , f(1,e), fle,1), and fle,e)

Let fkx,y) =xly+ ylnx Find f(1,1) , f(1,e), fle,1), and fle,e)



Answers

Find $\operatorname{gcd}(92928,123552)$ and $\operatorname{lcm}(92928,123552)$ , and verify that $\operatorname{gcd}(92928,123552) \cdot \operatorname{lcm}(92928,123552)=$ 92928$\cdot 123552$ . [Hint: First find the prime factorizations of 92928 and 123552.1

Mhm. Master minimize dysfunction subject to these two constraints here. Again, these are hyper planes and Rx wise in space and this is well um it looks like basically like a tabloid In a in our in our four dimensional space here. If you can well you probably can't picture what that is. But anyway um level sets of these spheres. Now, let's see here we can uh automatic function here. G equals plus number one C one plus then to see two again. Now this is going to be a function of X, Y. Z. Number one and number two taking partials with respect to all of those variables. We get five equations and five unknowns that we get to X one plus Lander one plus two equals zero two, Y plus two, lambda one equals +02, Z plus number two equals zero. And then we have our two constraint equations back. Now these actually are all linear. So we can just basically do back substitution and solve and won't get one solution. And that solution is eight thirds eight thirds four thirds. Mhm. So X. Y. Z. And if we plug that into f here we get 16. And of course we can all see that that's going to be a minimum because that's it's going this is probably going to be something like a problem if you you can obviously solve for um Y and Z in terms of X and plugged them in there and I think you'll wind up with a fabulous. But so it's going to be upwards because you can see that this thing is always kind of upwards curvature, so this indeed will be our minimum.

So the question is gonna look a little bit different. We instead of having one constraints, we're gonna find extreme value. So for question 38 we're looking to minimize the function F, which is given as f of X y z is equal Thio X squared plus y squared plus C squared. And then now we are given the constraints Um, X plus two I was three z equal six and expose three way place nine z is equal to nine. So we are gonna be using equation to for this problem that is given in the book for those problems that have two constraints and that says we want to solve the greeting of f is equal to land The times the grating of G one plus mu times the Grady int g too. And again Do you want Angie to need be set zero? So, for my constraints, I wantto have g one equal thio X plus two. Why plus three z minus X equals zero. And then g two is going to be X plus three. Why plus nine z minus nine is equal to zero. So if my first step, I need a final ingredient of these three equations. So there any of f I'm going to get to X. I was too. Why J plus Jews, e k for the gravy Aunt of G one. I bus partial dirt of wise too. And Z is three. And then for the grating mg to again get one for partial with with attract X So you with respect to why and then nine with respect. Okay, so I want to set the grating and Beth equal to Lambda. Great Angie. One plus in you sums Grady out of you too. So here I am really adding a lambda breach of these and I'm you here. So that means that too all the coefficients of the same vector gonna be equal to another. So that means that two X is gonna equal to Lambda plus Mu too. Why is gonna equal to to land a plus three mu and then to Z is gonna equal to three Lambda plus you incident I want to solve for X, y and Z. I see if all these air really already close, I just need to divide out to you from all these. So we'll go ahead and change all of these to being comes a 1/2 in front doesn't know. I can put these values of X, y and Z into my constraints. And what this allows me to do is that now I can get a system of equations that's gonna mean two variables because my X, Y and Z year all in terms of land A and mu. So if I plug those into here, I'll have two equations to variables. You solve it as normal. And then, once I get those values of Lambda Mu, I can plug those in to my X, y and Z. So first plugging these values into my first equation Here I get it. 1/2 slammed opposing you. It's my ex plus two times why? So I have two times what happens. Just one. So I just have to lam dead plus three in you plus three times e. So three halves, three lambda plus not in you minus six is equal to zero. And so when you simplify all this out and I like all your like terms, you should get that seven Lambda plus 17 mu is equal to six now, when do the same thing, but I want to move on to G too. So plugging in for X, I just have again 1/2 lambda purpose. You plus now have three times y so I'm gonna have three halves to lambda plus treating you plus nine z So nine halves, three land up plus nine. You minus nine is equal to zero. And then when you simplify this one out, you should get 34 Lamba plus 91 you is equal to nine. So now we have two different equations here. And so when you solve that system of equations for Lambda and you, you should get that Lamba is equal to 2 40 over 59 and I'm you is equal to negative 78 over 59. And so now I have the values of Lambda Mu. I'm gonna go ahead and plug that into my values for X, Y and Z. And when you plug those in, you should get The X is equal to 81 over 59. Why is equal to 1 23 over 59 and Z is equal to nine over 59 And so the minimum value dysfunction uh, this point is gonna equal to 369 over 59

Okay, So trying to minimize the function f of X y z given are too constraints. G avoid NGS to So in order to do this down of f mostly equal toe land A time of dell of G one plus mu times del of g two. So that means two X I had post to why j a happ was to the K hat must be equal to Lander. I had pushed three lander J plus nine and que That was del G one waas Oh, that's my mistake. That should be two and three and as ideology or plus new of I had plus three you jihad Plus no I you que that's, uh, g two. So when you combine a components, we get two x is going to be able to and a plus from you two, Why is gonna be equal to to lander plus three view to z is gonna be equal to to be lender plus nine. You okay, so now I'm gonna solve for the variables X y z so extra can be your friend of plus you over to why is gonna be able to to land a plus tree mu over to Z is gonna be good to treat when a person nine you over to. So given all these, we could plug that into our G one and G two equations and solve for that because that will leave us with two variables to equations. Something that's solvable. So G one, we have X So just gonna be handle plus mu everything will be able to a bodily and then to I So it's gonna be two times two lander. So push for Amanda plus six meal and then three z. So it's gonna be plus nine random plus 19 mysteries 27 you that's gonna be equal to six. And then we also have G to where we have x again. So just learned a plus you Everything's over to, um so that landed pushing you over to put three. Why so to Tom's story is six lander for a street machine. I You plus three times nine is gonna be 27 lander plus nine times nine 81. You that's gonna be equal tonight. Okay, so now it's adding all together. So you have nine landau plus four plus one doesn't be 14 lander plus seven plus 27 that's gonna be were, actually, yeah. 34 mu over 20 We go to six, and then on this side we have there. 600 person lander was 27 So I settle. Plus 27 again. 34. My under course nine puts 81 is 90 plus one 91 real and ask really over two is equal to nine. So let's solve one variable in terms of the other. Um, Well, actually, versus let's reduce this on the race, that seven Meet Lander plus 17 mu is gonna be equal to three. So I could send the salt for one variable, for instance. We could solve for the land if you wanted to. So, seven Landers and we go to three minus 17. You, Rand is gonna be equal to three months. Seven team you over seven. Good again for this side over here. So two times I was going to 18 So we're gonna have 34 Lambda plus 91 of you is gonna be 18. Then we can subtract over the view. So there are 34. Lambda is really equal to 18 minus 90 line. You lander is gonna be equal 18 minus 91. You over 34. Okay, so next we can set these two Landers equal to each other. So we're gonna have 33 months. 17 view is gonna be quick to seven. Over seven is going to be quick to 18 minus 91. Me over 34. Okay. So we can cross, multiply, and that would give us 34 ST will do US 100 to minus 5 78 mule. That is going to be equal to 1 26 minus 637. You okay? So now we can on super fire some things out. So, for instance, if we add 6 37 to the other side, we'll get 59. Mu is gonna be equal to 24. Correct? And then mu isn't Beagle 24 59. So I'm used 24/59. We cannot solve Lambda using either of these equations. So Lander is gonna be equal to three minus 17 times 24/59. Whoever said so asking to be you could took three months for over a over 59. I could reduce the three to well, 12 to 4 over 59. Oh, actually, that's gonna be 1 77 over 59 on a reception. So that will reduce to negative 231 or 59. That's absolutely divided lifesaving. So could, you know, time saving on So that is gonna be equal to Lambda is gonna be there equal to *** 33/59. So now that we have that because Sulphur X y z given our lander in our view, so X is gonna be equal to NE X is gonna equal to me percent over two. So that will be on that over two. Or but she could deny over 59 times two. So being actually equal to basically, this is an ignorant nine bodily negative 9/1 1 eight. Why would there be two times in 19 33/59 plus three times 24/59. All of to what wise that I'm gonna be equal to 6/1 18 which is all secret to 3/59 z is gonna be three times negative. 33 or 59 plus nine off 24 59 over to and then for Z. We're gonna get 117 were wondering to eat coordinates, therefore are gonna be negative. 9/1 18 3/59. 117

Today we are going to solve For the number 16 here we have to find X y and that that minimize it's way plus except minus two ways it subject to the constraint X plus y present equals two function. If at X y that Lambda is explain plus exit minus two, why's that? Plus Lambda into X plus y? Plus that minus two door By the way X equals Why plus that mine Islam Die called zero. Let it be question one love. But the way because explain this, Does it slammed equals zero. That would be question, too. The left. But those that equals X minus the weight plus lambda equals zero that would be three. Don't bite Dolan Diaco's X plus wife plus and minus two equals zero. That would be equation for so on one weekend. Right, Lambda Cause minus way Mine is that that every fight from two week ended lamb because minus X plus twos, it noted basics. So from three weekend right Lambda equals minus X plus two y let it be seven. So minus X plus two equals minus X plus tau white that because what let it be able from five domestics vigor, actually, because for is it, man? So from 784 bigger for the That's it. That's that equals to two. The deck was won by three from nine. We get executes for by three. From it, we get why he was won by three. Thank you.


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