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Use the mcthod of Lagrangc Multiplicrs to find the absolute maxim and abso lutc minimum of f(T,y,2) = 22 +y + 2 snbject to 12 + 2y? + 22 = 1....

Question

Use the mcthod of Lagrangc Multiplicrs to find the absolute maxim and abso lutc minimum of f(T,y,2) = 22 +y + 2 snbject to 12 + 2y? + 22 = 1.

Use the mcthod of Lagrangc Multiplicrs to find the absolute maxim and abso lutc minimum of f(T,y,2) = 22 +y + 2 snbject to 12 + 2y? + 22 = 1.



Answers

Find the relative maximum and minimum values. $$f(x, y)=x^{2}+2 x y+2 y^{2}-6 y+2$$

I'm ignorant as inequalities here, I'm gonna find a minimum value of two X plus three bites. Let us work on this inequalities. We take the equation X-us four. Wind equals age. I want to be here, X and Y. So now lives at stake here. Let's say access. It'll why is too we take no access age then why is coming out to be zero to The zero comma Through here we have. And the next point here we have that is 8:00 PM joined them joined two points or a straight line. And the question is Stephen, S. Or X plus Y equals two to B. M. X. And Y. Just take care access zero. Why is coming up to be drill? And if you take access that's the three wise coming out to be zero. So three comma zero here we have this is three and next to zero comma drills of true 4, 6, 8, 10 and 12. We have 12 to we just joined us. Where does line now? So now it is greater than equal to eight and this is greater than equal to drill. So we have to the physical region outside and there's a physical regional outside. So we have the physical region here. We have. Hello, this is an expert on ecological Weybridge and ecological the corner points we have that has given us on this eight comma zero second is we have this corner point International. This two lines services see here the intersection of these two lines. So we have explores. All right equals age. Next we have this is multiplied here. Uh, this was 4.2016. X Plus four. wife equals 48. So let's we have now Now there's no subtraction here. So we get negative 15 x equal to be a negative 40. Well as all this we have X equals 14 or 15. Eight or three here here now so we get eight or three. And next we have Report The value of now. Access it over three equally. We have 8/3. So 8/3 Plus four White as equal weight from here. You know four Y equals eight negative 8/3. That's come out of 24 Negative Age over three years old is now before right equals six years. We have 16/3. So why we have equals moreover treat. So you got one years for over three and access it over three. So 8/3 common for over three. The next corner point we have that has given us the last corner point is zero comma I'm gonna to minimize this. Two X plus three. Wife. You minimize this so we gotta dance the 16th then this will be get over 3. 4/2. That's going out to be two times 16/3. Plus we have well so on that gifts 16 positive. 12/3. So that gives continued over trick Next year. OK material that gives 36 so we have 16 20 28/3 4 million valentina. Look at the options 28/3. That is option one. So we go with their option one. Thank you.

We have been asked to maximize G equal 60 Next plus 14 were subject to Constraints X created an equal to zero by greater land equal to zero three x plus two wireless an equal 12 on seven x plus five by class and equal 29. Once we applaud the craft for these inequalities, this is what we get on in the solution said. That's the dark agree Nadiya. We locate our point. The first days zero comma 5.8. The second is to calm a tree on the third point as four comma zero. Now we maximize and minimize G By putting in these coordinates set, the four says, you know comma 5.82 comma three and four comma zero. We get the value of G for the first as 81.2 for the 2nd 74 on for the turd as 64 received the maximum and minimum from the street built and no doubt that the maximum of 31 point toe zero comma, 5.8 on minimum as 64 at full armadillo

The I'm in distance from the point to negative 11 two plain X plus y minus. Izzy is equal to so first tohave the distance formula and brought into here We're looking for them in mice in the distance, but formal is gonna be Do you have X Y Z is equal to the square root ovens around this point. Someone have x minus you squared. Plus why minus negative one, Which is why plus one squared Plus is the minus once word. So in order to minimize this function, D I would go ahead and just minimize the inside, so it's gonna be a little bit easier. So great. Careful the function and have equal to the inside of this radical and then to bring in this plane. I'm gonna go ahead and substitute in this disease, so I just have all of my X and Y variables looking up here. That means solving Proceed, A Z is equal to you. X plus y minus two. So then I get the function d of X y Siegel thio X minus two squared. Plus why plus one squared Busted. Now substituting in this value for Z, I get X plus y minus two minus one, which is minus three sway. So have its function. I want to minimize it. So we're fine, then in a month minimum. So this is gonna go back to what we've done in the previous problems in this section. So I'm gonna go ahead and find my critical point of dysfunction, and then I'm gonna use my second derivative test. Define the minimum. So to find my critical point first and you take the partial derivative with respect to X and Y is that the vehicle is zero and figure out which X and Y satisfies both equations. So first, taking this partial derivative with respect to X For this first part, I get just using channel two X minus two, and times wrote inside was just one. The second term goes to zero and a rose last dream. I get this similar thing X plus y story for D. Y's gonna look the same thing, but with why, so here. I'm gonna get a two times y plus one. I know how the same thing here and something both equal to zero. So I see both of these are the same term that means that these have to be equal to one another. And if you were to solve this system of equations, you should get that, um X equals 8/3 y equals negative 1/3. So we have the critical point. A third's a negative 1/3 and so poorly and eight. There's negative 1/3 into here. You should get that Z is an equal to 1/3. Okay, so Efron are critical point. Now we're ready for a second derivative test, and that is tells us that we want to focus on the value of this expression. In particular, we want to see whether it's positive, negative or equal to zero. In our case, you want this to be positive, and we want the second partial derivative with respect to X three Positive as well. So taking the second partial derivative here I'll get from this term. I'll just get a two from the stream. I'll also just get it to you. So two plus two is he cool? 24 go into why? I see. I'll get the same exact thing my wife's gonna have coached coefficient to same here. So I get two plus two. You again which is four and then looking at my 1st 1 and not taking it with respect to why this first time's gonna go to zero and then here I'm just going to get it too. And so, Phil, you must end up here. I get four times four minus two squared, which equals 12 which is greater than zero, which is exactly what we wanted. Also, my second partial derivative with respect X is greater than zero. So therefore, by the second driven a test, we have a local men, as now plugging in my point, have hey thirds negative 1/3. And we found Ze to be 1/3 and putting that into my function. Here I get that this comes out to you two over the square root of three. And so that Fineman Mom distance

To be able to find out absolutely maximilian. That's been long early of the grandparents and gov heads That it becomes to access 1-plus who went to X. To the power to wait three on the given table. one is to you. Okay. So we have to find Absolutely that's one you must first we're gonna get this uh that is a surprise of X plus forward three extra. Repower minus three. Yeah. So it is six h. to the power 413 plus four upon. Okay. We have to live around one by three. If it is a question zero, he does have a question zero. So on here Actually. Power of four x 3. That is the question one is 4 x six. It's not possible. Yeah. And next 25 that was him for that. So now we close to zero. So you can see that you can see that at express to zero. The media sex not defined. So it is only only critical one. Number one. Yeah, employees managed to the group. And then we managed mm mm. Now we will find the value of the function at Columbia. And then point so GO -2 energy of judo and G. Of school. Yeah. So do you all minister baby man. S. P. Squared plus two in two months. To the power of three. We start less queen our poetry. And it will be questioned mm hmm. So it will be questioned my school. Much to the power of who was he? Mhm. 22. Mhm. This four. So can you just the same point more than seven. Okay. Bye. So you can see that from the head. Actually make some value. Yeah, absolutely met. Some value is 7.175 and absolute minimum right value is zero. So, these are an answer. I hope you mr Thank you.


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