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Use tne given teasidie egion determined constraint Inequalties objective function (if they exist). (If an answer does not exist, enter DNE:maximumtne givenIio. 3)ma...

Question

Use tne given teasidie egion determined constraint Inequalties objective function (if they exist). (If an answer does not exist, enter DNE:maximumtne givenIio. 3)maximumminimumNeed Holp?endl

Use tne given teasidie egion determined constraint Inequalties objective function (if they exist). (If an answer does not exist, enter DNE: maximum tne given Iio. 3) maximum minimum Need Holp? endl



Answers

Find the minimum and maximum,, if they exist, of the objective function $f,$ subject to the constraints.
Objective function: $f=4 x+3 y$
Constraints:
$\begin{aligned} x+y & \leq 80 \\ x-2 y & \leq 0 \\ x \geq 0, y & \geq 0 \end{aligned}$

So we have to find a maximum and the minimum for you for the objective function which is zis because three X. That's why and it's subjected to the feeling going strange to have X. Is greater than I go to zero, why is greater than we go to zero. You know three eggs. That's why it's less than equal to 15. We have four X plus three Y. Is less than equal to 30. So these are the constraint. Mhm. Yeah. So now let's try and put the constraints. So protein, it's you're going to get something like this. Mhm. Yeah it is Okay here at 50. Okay and here it's 15. So you get 0 15 and the next time is here Yeah You're going to get 7.50 over here. So we can see an intersectional case over here. Hold A. So now we have to find the coordinate of the point where the two lines intersect and these are the 2 93 X plus Y is equal to 15. This is the equation on Yeah. Four X plus three Y equals to 34. Asian too two from the equation one, you can make exit subject to equal to five million S. Y. 43 from the question to we get X 24 to 7.5 minus 3/4 Y. So are the intersection at the intersection point in the X. Values are the same. So we can equate to question three and four. So we get Y to be equal to six and substituting and Y. You got the six into the equation three We get X two weeks. So we have the coordinates of intersection to be 36. So when I began to find them. So at four practices of the written by the mainstream the objective function has different so we know that observe reserves So that there there we know this, it was three. X plus Y. So substitution and probably get three times there because uh two weekends there over here. So at 50 You can see to be called three times 5 Because which is equal to 15. So at 010 get Z is equal to three times sarah. The last thing I just was with them. Yeah at 36 Again C. is equal to three times 3 1st six Which is equal to 15. So looking at all these values for the four practices we get and that the maximum value okay to market value of the sea of is 15 and it occurs at the port needs 36 and five and fives there And the minimum value is zero. So minimum value is there and it's okay at the point ecstasy go to zero and y is you go to zero

So we need to find a maximum and minimum values for the given objective function is subjected to the given constraints. So we have, you have to find a maximum and valley for this objective function. We have C. Is equal to express why. And the constraints are you have X. It's better than their or equal to their have greater than equal teacher. You have three x. That's why it's less than 24- 15. You have four X plus three way is greater than not, it's light than I got two, etc. So these are the constraints. And let's try and grow this. Mhm. So girlfriend is the area determined by the constraints. Has shown us. So you're going to get something like this? It's just a sketch. Yeah. So they say you have to turn over here and let's say we have 15 somewhere over here. So we are going to get here to be 015. We have here to be 010. We have here to B. There is there we have here to be five. So and I see here is 7.50. So this line is three X plus y is equal to 15. We have four X plus three. Y is equal to 30. So now we have to find the coordinates of the point why the two lines intersect. So we have three X plus three. Y is equal to 15 to 6. To go to the equation. We have four X plus three Y is equal to 30. Secret equations. So from equation and we're going to get why excessive go to five minutes white factory We live with a question three. And from equation to you're going to get X two we go to 7.5 minutes three about four what this is equation for. Mhm. And then so at the intersection the Point X values are the same. So we can equate it machine train wishing for to get white because 36 And substituting the value into the equation three you're going to get XTV. I'm positive three. So the coordinates for the intersection is 36. Yeah. So now at the four practices of the region formed by the constraint the objective function has the following valleys. We have 00 so at 00 we are going to get Z to be called to zero class zero which is zero and this is a minimum valley fancy and we have 50. So advisor we are going to get seen to be able to 50 which is five. Yeah The next one is everything. Yeah. So you're going to get Z to refer to zero plus 10 Which is 10 and 10 is a maximum value for Z. And you have the last time to be 36 so 36 you're going to get Z to be called to three last six which is the No so now we can make a conclusion. The maximum value of C. Yes it was within and it's okay that um Ecstasy go to zero and why is people take them? This is for the maximum value obviously, and for the minimum value of C. Is equal to zero. And it's okay at X is equal to zero and Y is equal to zero.

So we have to find the maximum and the minimum value. And in this question they were given the objective function to be equal to Z. is equal to two x plus y. And this objective function is subjected to the clinic on strings. You have X greater than zero. Why is greater than zero? You have three X plus Y. Is less than or equal to 15. And you have four Y plus three Y. It's less than or equal to 30. So these are the constraint. Mhm. So the area determined by the constraints are shown below. So I'm going to show you the area. So now we have You have to find the coordinates of the point where two lies intersect. So you're going to get three X plus why we go to 15. This is equation one. Yeah. Four eggs Plus three by close to 30. This equation too. So from equation one we are going to get X to 35 million S. Y over three. And from equations we are going to get X. two because 7 very minus three. The full boy. So And the intersection point of X values are same. So we can equip three and 4 And we get YR six. So substituting this value into the equation you're going to get eggs to be equal to three. So the coordinates of the intersection point will be three and 6. So at the four victims is of the region fund. By the constraints the objective function has the following values. So the first one we have at 00. They're going to get Z to the ego. 2 to 0 cleanser. Mhm. This is zero and there is a minimum minimum value of C. So the next one is at 50. They're going to get it to be two times five plus zero which is equal to 10 mm. So we have a 010. I think he gets it because the two times there last thing this is equal to 10 and the last point we have at 36. Mhm. It ain't easy The to be equal to two times three plus six and this will be equal as you top. Yeah So 12 years a minimum value obviously. So the maximum value so let me right max the maximum body of the is tool And it's okay that exploded three. And why is it was 6? The minimum value It is Z is equal to zero. Case at X. is equal to zero and why is equal to their

Thanks we need constraint mm This is a have access greater than equal the advice the three x plus y. See Bean all yeah place I mean we need X intercepted by at accent. Why is it So three exes gives us see right excellent. Five the then next we need y intercept X zero that's right team get vice venus wind so this is us by intercept zero comma mm If we join these wayne we have uh yeah We have three x plus y. Less than equal. Less than me. We need to no the list. So this is a Sure now next we have ford Explorer the right an equal graph in it. For express everybody is equal. Then we need eggs in mind this so right well you X is equal. I mean by point place invite to coma Then we have three wise equal mm give the sake Y is equal. I want to say coma. Yeah. And we need to join these. What? Uh huh. One X less the right an equal the so it is all. No the common area will be this area region. Minister common mm four corner points we need the point of intersection of these two lines. We have Y. Is equal being minus Yes. So let yeah four X plus three times being Linus Yeah is equal. We have four X plus 5 -9 X is equal. Mhm. So minus by effects is equal minus the and mhm So x coordinate is three and Y is equal Been -3 times three that there's 15 -9 and why is equal? So our corner points are zero comma 10, three comma six zero comma zero and five. Uh huh. The next we need the maximum and minimum values. Let us see the graph graphing calculator for this. Yeah object function is equal right less right and mhm. Region is form and shape. Chin corner points three comma six. The needle. My mm zero comma zero high school zero. Now let us see the value all of these corner points. So that uh zero comma yes zero plus 10. Yes three comma six. We have five times 30. I mean yes. What 21 five comma we have doing Last zero. That's doing right. And at zero comma zero we have zero plus zero right zero. The minimum value Euro commas oh and the minimum lewis go on. And my Value lies at 5:00. And my. Yeah green. That


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