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Find the maximum and minimum values of $f(x, y, z)=4-z$ on the ellipse formed by the intersection of the cylinder $x^{2}+y^{2}=8$ and the plane $x+y+z=1$....

Question

Find the maximum and minimum values of $f(x, y, z)=4-z$ on the ellipse formed by the intersection of the cylinder $x^{2}+y^{2}=8$ and the plane $x+y+z=1$.

Find the maximum and minimum values of $f(x, y, z)=4-z$ on the ellipse formed by the intersection of the cylinder $x^{2}+y^{2}=8$ and the plane $x+y+z=1$.



Answers

The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 + y^2 $ in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

The traveler. Mr. Plane for X minus three y plus eight days ago Too far intersects the corn they squared is equal to x squared plus y squared Emmanuel lips. Part a raft, The corn hands a playing on DH observes electric or intersection. The little cat is graph. Here the plane deflect the corn united lips here. Is that his lips hard to be used like arrangement players to find it the highest and the lowest points on their lips? Well, Poppy, we need to find is a maximum mario on many my minimal while you see that is able to think with constraints or fax minus three. Wife asked. Thie, You come too far. Yeah, we didn't know that this is she. It's This is you got X Square plus y square. One stay square is equal to zero with two strains. Then we have ridden off ofthe safe. Or to love that ham spreading tough d for us, Mew towns. Gradient of it. Gee, is they control? Half inch is equal to zero, which become zero. You caught you Orlando plus two packs. Zero is equal to negative. Three landa us too mu. Why? One, two eight Lunda, minus two mules. See, relax minus three wine. Plus it. Thie. It's the little father, and it's a square. It's equal to X squared us. Why score So from the two first of three questions we have lumberyard hams. Why plus three acts? Is the culture zero through? Not that Lambda is not equal to zero, because otherwise, from the Question three we have mu is not equal to zero. And from the first two equations have access because why it's equal to zero. But and from the last equation, C is equal to zero, which is country. Which contradicts was question for London is nothing with zero. So we have four y plus reacts. It's equal to zero. Then we solve the questions Well, why plus three AKs You got zero or axe minus three. One. It is the You go too far. On that square is cultural X squared plus y squared. We have X Y Z. It's the culture negative or homer. Three. One off over three and or or thirteen collective three over thirteen, half over thirteen. Then we have the highest points on the lips. Is this one on the lowest point on the lips? Is this one

We're trying to find the closest to the origin of the intersections of plain to Why plus or C five and four times X squared plus y square decrying Z square, the distant function used with the FX. Why common C, which is equal to X squared plus y squared plus z squared She won, uh, X comma by common received Be two y plus foresee Brings you to look X corn Ex con y common Stevie Plus Why, Where things multiply Backboard Mine is C squared greeting a path from two x i plus two y j had plus two. Okay, hot grating of G wanted beef zero I plus two j plus 4/2 greeting to would be equal to e x I a high J minus two. You act equal to you multiplied by E X Um, you be equal to one warder or ex implying hero following that too. Wise chill lander for us e times why you way have to seize the foot four land uh, miners to see Neil. If X is news equal to 1/4 uh, Lambda supposed be equal to zero limit will be equal zero and Havana, if land isn't there into the equation. You'll be Q C minus my yes, you half Z. And by the way, that I eat. Quote. Unless unless he's equal zero. But it's not. Not possible. So we use X equals zero. So X equals zero. Your place X equals. We use two. Why? It was for sees equal to five and then four times why square e claims sees square. See with then be called a plus or minus to replace e into the equation here and me and wide five war my ass. Six wise equal to five while legal half or why equal minus five out of six would see here Equalling zero Hubie equal one. And the other one would bc. Okay, 10 out of 10 out of six. Which is I out of Marie. So then the two equations points are your, uh, kind of one or or zero comma minus five out of six comma, five out of three off your okay. Uh, common one is equal to five more of zero column. Negative five comma 53 which is equal to on 100 100. High, divided by 36 which is greater than 54 So therefore, the the point. The point closest is, uh, you to the Oregon. I mean, Euro common Come one

Using a grand multipliers, we get two X equals. I am, uh, Times four execute and two Why equals land? Uh, times four y cubed and to Z equals Lambda Times four. Is he cute? Now this first equation is going to give solutions of X equals zero our land. Sorry. One equals William. Uh, Times two x squared. The second equation will give y equals zero or one equals Liam toe times two y squared and the third. Likewise, Z equals zero or a one equals Lambda Times two z squared. Now, if we look at each of these equations green, we have Lambda Times two X squared is equal toe NMDA times two y squared Sequent, Ill. And attempts to z squared We're all equal to one. And so if we just divide each of these by two lander we get that X squared because y squared you could see squared. And for the constraints equation, that means we have by substitution X squared, squared, Paszek squared, squared plus X squared square equals one which, of course, gives three X to the fourth equals one and so we get X equals for the fourth root of 1/3 and likewise for Y and Z. And so if we point this in two F regardless of whether they're positive or negative, we're going to end up with Route three. And no, if we look at X equals zero, for example, if X equals zero, then we get mine is the fourth proceeded. The fourth equals one. And so let's say why it was tea than to satisfy this constraint. Z must be the fourth root of one minus yeah, t to the fourth. And likewise for Y equals zero and Z equals zero. We can introduce the points t zero fourth root of one minus T to the fourth and key, the fourth root of one minus t to the fourth zero in each of these are going to give a value of one one point in it a function. And since one, it's less than Route three, these will be our minimum. And this is the maximum

If we usual grounds multipliers. We get four x cubed Hagel's landed times two x four White huge equals Lambda Times To why and for Z cute equals Lambda Times to Z, and that's going to give us two solutions for each equation. So for the 1st 1 X equals zero, our Lambda equals two X squared y equals zero or Lambda equals two y squared for the second equation and the vast or you the equals zero or I am the equals two z squared. And if we take all these William the equations and use them to say each of these equal to each other, we have two x squared. It was two y squared because two z squared, which of course, gives X equals bite ze. And if we use this to substitute into the constraint equation, we get X squared plus X squared. Plus X squared equals one, which gives us three X squared equals one or X equals plus or minus one over Route three, which is equal to Y and Z as well. And so if we just plug in to the function, we'll have poster murders one over Route three as the values for X, y and Z, and it really doesn't matter whether each of these air positive or negative, since each x, y and Z in the function is raised to uneven power. And so, regardless we end up with one night was one night was one night, which is three nights or 1/3. So there's one value. Now, if we look at the solutions, X equals zero y equals zero is equal. Zero. We'll take X equals zero, for example, on that shows us that y squared plus C squared equals one in the constraint equation. And so if we have Y equals some value T, for example, that means that Z has to be equal to the square root of one minus t squared. And since the square root we can't take the square root of a negative number. One minus t squared must be greater than or equal to zero. So we see that T squared has to be greater than or equal to one and said T has to be between negative one and one. If t is between negative 11 then the scaler scaler value of T is going to be through or a fraction and neither of the's one point into the function. We're going to give a maximum value. And so we see that we were we'll take to you goes plus or minus one, and that will give us points that look like zero plus or minus one and zero. And likewise, we could have poster minus 100 where 00 poster minus one. And all of these one point into the fashion. We're going to give us one and one is greater than 1/3. So this is our maximum value, and 1/3 is our minimum value.


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