5

Find the minimum and maximum values of the function f(r,y,2) = 3r + 2y + 4z subject to the constraint 12 + 2y? + 622 = 16fmarfmin...

Question

Find the minimum and maximum values of the function f(r,y,2) = 3r + 2y + 4z subject to the constraint 12 + 2y? + 622 = 16fmarfmin

Find the minimum and maximum values of the function f(r,y,2) = 3r + 2y + 4z subject to the constraint 12 + 2y? + 622 = 16 fmar fmin



Answers

Use Lagrange multipliers to find the maximum and
minimum values of the function subject to the given constraint.
$$
f(x, y, z)=x y z ; \quad x^{2}+2 y^{2}+3 z^{2}=6
$$

If we use the grunge multipliers, we get two X equals by 100 times. One to why equals Lambda Times one and to Z equals Lambda Times one which, of course, gives us two x equals two y equals two z Since they're all equal to Lambda, which implies X equals y equals e. So if we use this information in the constraint of X plus y plus Z equals 12 we can say that X plus X plus X equals 12. And so three X equals 12 and X equals four. Which, of course, means that like was four and Z goes for as well. So now we plug in for the 0.444 into our function, which is going to give us four squared plus four squared Plus four squared, which equals three times 16 or 48. Now to figure out if this extreme value is a max or min, we're going to have to test some other point that satisfies to constrain. So the easiest thing to do would be to let X and y both be zero and then Z, of course, has to be 12 so that there's some equals 12 And if we put that into the function, we get zero squared plus cereal squared plus 12 squared, which is 1 44 which is greater than 48. So 48 can't be a max, so we must have a minimum of 48 at 444

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

The function we look at here is F X Y C. It's echoed too. X square passed to y square as she T square into subject to explains to my person city, You go to 10. This we are going to find a minimal value. You wanted to find a minimum level. So we're going to use one course policy throughout the inequality. This equality is telling you say the A squared plus B squared plus C script James X square plus Y squared plus the square. It's bigger than a X plus B. Y plus C C square. So how do you apply into this problem into USC one is x square plus two. Wide square plus. See the square and you also times one plus two plus three and it's well bigger than express to I pass sweetie square. So this means this is F X Y lee And this is a constant. We already know that is 10 into this or already we can compute, So that means F X Y z is bigger than 10 square over six. That is 50/3. And you wanted to attend. Uh in order to attend the equality. So the equality attends Is when this x square over one is Echo two. Two square over one is equal to three. square over three. So that means mm X echo gy echo to see echo to analytics or fire suit wow. So the second one, we're going to show this F has no maximum values with respect to this constant. And to do this, we can simply choose the say the constraint here Is expressed to one per sweetie equal to 10. And here we can just to pick pick some data points. Say, see, this is 10 minus K, two K in this K in this zero. So this live this metaphor are to this satisfies this condition. Okay, So then you're going to compute f x Y Z. That is 10 -2 K sq. Because two k square class zero into this means so, yeah, yeah, you can see. So this is at least bigger than two k squared. Right? So, because this is always bigger than zero, and we can find the lower branches to k square. So this means when K tends to infinity into two K square, where it goes to infinity, and F f x Y Z. Where also goes to infinity. Because it's a lower bound s infinity. And this means F has no maximum values with this constraint

If we use the grunge multipliers begin why e to the X Y equals land times three x squared and we get X E to the X y equals land. Uh, times three y squared it's in. The next step is going to be to multiply both sides of the first equation by X, and that will give X y E to the X Y times three x cubed and then for the second equation will multiply both sides by why. And that's going to give us X y E to the X Y on both sides on both left sides of the equation. And so we can say that I am the Times three x cubed is equal till the end times three y cubed, which, of course, gives us this equality where Lam does cancel and three is cancel. So we have X cubed equals y cubed, which gives us that X equals y. And so now we can plug into Did you quit? The constraint equation, which was X cubed plus y cubed equals 16 and we'll just plug in expert. Why? So we get two x cubed equal 16 which gives X cubed equals eight and so X equals two. And since X equals y, that means why it was, too, as well. So now we just want to evaluate the function at 22 and that's going to be e to the two times two or into the fourth. So we know that effort to two equals either. The fourth gives us an extreme value, but we don't know if this is a max. Where men on easy way to figure out whether it's a Mac sermon is to just look at some other point that satisfies different strains. And it's a part of the function. So the easiest way to do that is to test X equals zero, and we see that X equals zero on by the given constrain Give zero Q plus y cute, he calls 16 and so why must have to be the cube root of 16. And if we put that into the function, we get he to the zero times the cube root of 16 which, of course, is just eat to the zero, which gives us one and one is less than either the fourth. So we conclude that either the fourth can't be a minimum, and therefore it must be maximum. So we have a max of eating 1/4 at to to


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