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Maximize 58 +x + xy 5y7 , subject to the constraint ~ 6 -X- Zy = 0. The maximum value of 58 + x2 + xy ~ 5y? subject to the constraint ~ 6 - X ~ Zy =0 is (Type an ex...

Question

Maximize 58 +x + xy 5y7 , subject to the constraint ~ 6 -X- Zy = 0. The maximum value of 58 + x2 + xy ~ 5y? subject to the constraint ~ 6 - X ~ Zy =0 is (Type an exact answer in simplified form )

Maximize 58 +x + xy 5y7 , subject to the constraint ~ 6 -X- Zy = 0. The maximum value of 58 + x2 + xy ~ 5y? subject to the constraint ~ 6 - X ~ Zy =0 is (Type an exact answer in simplified form )



Answers

Find the maximum value of the objective function $f(x, y)=8 x+5 y$ given the constraints shown. $$\left\{\begin{array}{l}x+2 y \leq 6 \\ 3 x+y \leq 8 \\ x \geq 0 \\ y \geq 0\end{array}\right.$$

So we're gonna be given an objective function f of X and why that's gonna be equal to eight X plus five. Why? To want optimized or maximize dysfunction. And it's that it's gonna be released to do it without any constraints. So we're gonna have some constraints. These constraints were going to come in the form of inequalities. The 1st 1 is gonna be two x plus. Why is less than or equal do seven x plus two. Why is less than or equal to five? Then we have the two and qualities that constrain us to the first quadrant, which are X are great, is greater than equal to zero. Why is greater than or equal to zero? These values here just constrain us to real numbers or positive numbers, including zero on these to set the other properties. So let's go through. And let's graph Thies, too, inequalities out on a graphing calculators. But first we gotta get into y equals MX would be format career. We just attract both sides by two. We get why it's less than or equal to negative two X plus seven, and here we have to subtract by accident. Bible says But I do speak why it's less than or equal to neither 1/2 X plus five, Hafs. Okay, so let's go through. Let's Grafite. So contract with the 1st 1 which is fly is less than or equal to night of two X plus seven. Okay, that gives us our first region. Right then we have X book. Sorry. By minus 1/2 X waas five Hafs. Oh, sorry. I should be a less than equal to right. That's what we want. Then we have, of course, X is greater than zero. And I is greater than or equal to zero. So I'm gonna unmarked these because we know we're working The first quadrant just makes it easier to see However we know that we have to have. But our solution is within this region within the solution region right here or the feasible region. Okay, Mark bills Just just so you don't forget within this feasible region. And we also know that it's going to be one of the, um, one of these points here. Okay. Could be 00 It could be 3.50 It could be 1.8 comma, 3.4 or it could be. Zero comma, 2.5 could be any of these points. So the way we're going to do that is by testing each point. Let's go through. No, erase this. We don't really need these anymore. These were just or constraints. They told us they told us where we could work. But now we have where we can work. We want to figure out what values those would give us. Okay, so we have 00 from it, like no zero that we have 2.5. They would have 3.50 Then we have 1.83 point four. Okay, so this gives us our set off points. That could be maximum. Right? So, you know, it's one of these points, but we don't know which one. Okay, so let's go through. Plug them in zero plus 00 Do you know times 80 in 2.5 times five is gonna be 12.5. So we get 12. 15 Here we have 3.5 times eight. It's 28 plus zero. What's 20 plus year or not? 280. And we have 1.8 times eight plus 3.4 times five. And that gives us 31.4. Okay, so we can see that this now here gives us the highest value of our objective function. So right, we drew these this graph so that we had these points here. These four points put them all in. We got 31.4 is gonna be the greatest off the objective function when we use the values 1.83 point four.

In this video, we are going to fighting maximum. Well, Lou, off this function you when restaurant to this car for X plus two y equals wealth. So we're gonna have function g as this one on Dhe. Since we I do sing the Legrand's multiply o methought we are looking at the radiance equation. First, I have calculate this out. This two is a bit unusual because we have fraction power, but you can do it Still, because it poorly no meal and comparing the i l j accorded it. Who came up? Give us this system. Lambda equals the I coordinate. Boo, Give us in over five. Oh, why? Over X power by one fit. And this is also from Jake ordinate equal to four or five times Explore Why Power four or five? So we compare this it do give us that, um to why Equals takes yes on. And we gonna use this to fi the value off x and y at the intersection. So we put it in this form. We're gonna have that since X is to y we put this in like this. So it 10. Why equal 12? So why is 12 over 10 ridges, six over five. And so Hey, he's two times that. So is 12 4 or five. Now we have the value off X and Y at the intersection. It is, uh, X three months. So we put this in, I'll function. Do equals it. Times 12 or five. This new power bi 45 and six over five. Power bi won over five. So, Eve, I would too be an exa miner. I will give this correct like I would I would give this fool school already. But if you worry, you can try to simplify this. First you see that below We have five power by forfeit and one fits. So it combined even give fire, right? And a bullet here we have until told, we have three to the 45 and two to the head. Five from the 1st 1 times tree to the 1/5 tons to to the 1/5 for the second terms. And so go up there. We have this to become does tree right? And this to combine will give one to leave leave to to the 4 50 inside. So I think this is the correct answer. But again does. Arriving at these step is arriving at this is enough for me, actually. Yeah. So this is the maximum Off you win. Restrict to this graph. Thank you.

Because I was trying to find the maximum to the following equation. These recorded three x. Last five Y. And we have the following uh constraints. They are. X is bigger than zero, so is why And negative X. That's why he's also created our fault. So for this we're seeing that we're having X and why positive here. So the more we can increase X and Y the better. So I'm having that X has to be positive. Why has to be positive? No problem. And also negative. X plus Y should be bigger than four. Or you call to. Well, what if I say, why is equal to, for example, be plus four and X is B then minus X plus Y is four, therefore it's within the ranges and the constraints that we have. And if B is very large, they send the equation we have for the zero just increase. And since we can just make the increase forever, we don't have actually a maximum. So there's no lights.

We're trying to find the minimum for functions E is you call to 1/3 X -2/5 by And we have the following constraints we have six. Is the center will go to X plus Y. The center it looks wait and um for the last time equal to negative X plus Y. The central quarter to six. Now it becomes okay, add up to two constraints together. 6-plus 4 gives us 10 plus X plus Y minus X plus Y. Gives us Just to why? And 8-plus 6 gives us 14. Um Now though I did the numbers over by two gives us why? Let's listen our April two. Why the center april to seven? No, you can see that. The limits for X and Y. In our constraints, The concerns for them are one but in our function the constant for X is one third Now, for why is 2/5? Go over five is larger than 1/3. So why should be the priority first of all? So, let's try to Because you have a negative maximize why to get the minimum value for Z. All right. So what is maximum for why? It's seven By equals to 7? No. Considering dad. Let's try to find our minimum value possible for X. So for that, we need to subtract the first, the second equation from the first one that would give us Uh 6 -4 to It's less than or equal to X. Two X. A centre april 22 Oh yeah. So we have only one option for X. X. is equal to one. All right. So now let's find sleep genes equal to 1/3 X. You should just one third minus 2/5. Y. Because they have seven for radios 14/5 and the common denominator. We get 15 here, five here and -42 here and adding it all together, we get Neither. over 15. And that's how we're finally answered.


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