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Find Ihe absolule maximum and minimum if ctlher exrst, fuxt Ihc function on Ihe ichcaled interval {x) = (x 38x-7) (A) [0, 6] (B) [2 , 8] (C) [6, 9 (A) Find the abso...

Question

Find Ihe absolule maximum and minimum if ctlher exrst, fuxt Ihc function on Ihe ichcaled interval {x) = (x 38x-7) (A) [0, 6] (B) [2 , 8] (C) [6, 9 (A) Find the absolute maximum Select the correct choicc below and necessary; fll Ihe answer Loye: complete your chokce: 0A, The absolute maxmum Is Jx= (Use comma t0 separate answers a$ needed ) 0 B There no absolule maxmumFind Ihe absolute minimum Select the correct choicc below and, i necessary; fll in the answer boxes f complete your chcice 0A The

Find Ihe absolule maximum and minimum if ctlher exrst, fuxt Ihc function on Ihe ichcaled interval {x) = (x 38x-7) (A) [0, 6] (B) [2 , 8] (C) [6, 9 (A) Find the absolute maximum Select the correct choicc below and necessary; fll Ihe answer Loye: complete your chokce: 0A, The absolute maxmum Is Jx= (Use comma t0 separate answers a$ needed ) 0 B There no absolule maxmum Find Ihe absolute minimum Select the correct choicc below and, i necessary; fll in the answer boxes f complete your chcice 0A The absolute mininum alx= (Use comtma to separate answers as needed ) 0 B There absolute minimum; (8) Find Ihe absolule maximum . Select the correct choice below and, necessany Tmin Ihe answee bores complete your choice 0A The absolute maximum alx = (Use coninia separate ansiets 33 needed | 0 B Thete no absolule niakium Find {he absolute minimum Selectthe correcl choice below and necessary; fill in the answer boxe; comp ele your choice 0A The Jbsolute minlnum is JL<= (Use conima {o separale ansvers J$ needed ) C B Thcte no aLsolule minimun (C) Fina Ihe ebsofute maximum Se Ect Ihe correct cholce beljiy ano; [email protected] the Jnswe L0X65 comolel? Your Encce The absolute maxlmum i5 al = (Use conima (0 sepzralc 0nsi,213 needed There no Jbsolule marmul Find the absoluie min mum: Select tne correct choice /elowara @edeszan MIn tre anstve Dores comp cle Ycu choice The aosolule mirimun [ aix = (Use connnia sefaralc Jnst;ert J rceded EB. Thete no absclute Minimun



Answers

Finding Minimum and Maximum Values, find the minimum and maximum
values of the objective function and where they occur,
subject to the constraints $x \geq 0, y \geq 0,3 x+y \leq 15$
and $4 x+3 y \leq 30 .$
$$
z=3 x+y
$$

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

We first want to show that this function has a minimum. So we differentiated with respect to why every first back to eggs and with respect to why we didn't said the derivatives to zero. And we have the equation two x plus two y to be equals to one and that shows that there is a minimum or there is an interior minimum that lies on this function. So we next look at when excess equals zero and when excessive goes toe one because these are the it just so you want to know if this a critical point along the edges. So for excess equals zero, we set all x variable also beat you both zero We differentiated with respect to why said the derivative to zero and obtained a value off Why, which is half in this case We didn't find if the value off the function when existing also zero NYC also have and that use us three divided by four. We repeat this the process with excessive goes to one. So again we replace all ex variables with one differentiated by with respect to why find a value off why? And in this case, we have why? To be both a negative half which happens to be out off the range relative it so we can safely This caught this case we next to look at the other two edges when wise it goes to zero Follow buyer NYC Also one when y z equals zero We set a y ver both zero different stated with respect to X set the derivative 20 finally value off X wish in this case is half and we find that the value of the function when excessive also half and why's it was zero The very off the function at this point is three quotas and then why Citgo's zero We repeat same process and the value of X in this case is negative hop which is out off range. So we can also this caught this the critical point along this age last Me we look at the four corners so extreme just from 0 to 1 and wiring just from 01 So these are the four corners and we're looking at and we find that the absolute minimum of this function within within the prescribed limits Um, this three quotas and the maximum is three

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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