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Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,2) =x2+y2 + 22 subject to the constraint x +y +2 = 12 80,412)...

Question

Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,2) =x2+y2 + 22 subject to the constraint x +y +2 = 12 80,412)

Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,2) =x2+y2 + 22 subject to the constraint x +y +2 = 12 80,412)



Answers

Use Lagrange multipliers to find the maximum and
minimum values of the function subject to the given constraint.
$$
f(x, y, z)=x^{2}+y^{2}+z^{2} ; \quad x+y+z=12
$$

To optimize the objective function. Given f here we'll start by finding the partial derivative of F with respect to X, which is three and f with respect to why, which is negative. Two. And then we'll do the same thing. Fergie. The constraint function and the partial with respect to X. It's two x with respect to Why is or why and then from here will multiply each of the terms on the right by the Legrand multiplier. And then we're going to use the system of equations to come up with the relationship between X and y. So let's see. We could just go ahead and kind of set the top equation equal the bottom equation. It also we have three equals, two x times that Lambda. And so if we solved that, for instance, we just get three over two X equals lambda. And in the bottom we get negative to over four y equals lambda or negative on over to why, and then replace the Lambda from that first question. So basically, you know, setting up movable equations here from here we could cross multiply so we'll get sex. Well, I he calls negative two x and then dividing, we'll get negative. But really, why? Cool sex? And then this is the relationship that we're going to pull again to our constraint, which is the second equation here. And doing so, we'll end up having negative three. Why all square and then plus two y squared equals 44. And then we could see that this is going to get us nine. Why squared? Plus two Weisberger 11 y squared, dividing that over. So then that Give us why Square 44 divided by 11 would get us four and then finally we could do square root. But again, that would get us other plus or minus to And then from here we could substitute these terms back into the previous relationship and end up getting that X would end up being either positive or negative. Six. So now, looking closely at at the values given very positive negative to positive negative six, we want to see how they would do in this function. So which, which set would maximize them well, to maximize them, we'd want a positive X, but we'd actually want a negative. Why? To get that to become an addition. So that would be sex negative to would be our Max Max and then our men. On the other hand, we want the first term to be negative, and so we'll start off with negative sex. But we wanted to retain its attractive property so we'd actually want the why to be positive so that we'd end up subtracting for their And so that's it. You can always use the objective function toe, verify whether it's a maximum in, and you can even use the constraint function. Come up with your own set of variables that work in that equation and then compare them to the ones that you generate to see how they would do.

If we useful grunge multipliers than we have two X equals bland Why? And two why he calls land uh, ex. So if we multiply these equations together, we get four x y equals lander squared X y. And since X Y equals one, we have four equals Lander squared, which gives us Lambda equals two. And so, if we put this into two, X equals Linda Why we get two x equals two. Why? Where x equals y essence X y equals one. We have X squared equals one for X equals plus or minus one. And since X equals why we have the points 11 and negative one negative one which both give the value of two no sense f of x y is X squared plus y squared and X Y equals one does not provide an upper bound. There is no maximum, which means that these two must be minimums. So are finally answer is that we have I minimum of to at 11 and negative one negative one and we have no maximum

From Chapter 13. The problem gives the following equation F and says it's subject to the constraint given on the right. We need to find the LeBron multipliers to then find the maximum and minimum values for F if they exist. And that is, of course, subject to this getting constraint written on the right now were you rewrite that as G function and study equal to zero. So we have to do is just move the one over here Now we can do it's from the great of F and G Ingredient is finding the derivative of each function with respect teach independent variable. Now we have to find the LaGrange mold apart divisions Sweet said the partial derivatives equal to each other but multiply land times G's person derivatives in the last condition is writing G equals zero, which we did above. So now we need to rearrange these three equations to find X and went out Well, we can see right away is that this equation? But it has won a lot of science. We can just equal the right sides, and through some algebra we will discover that as equals y. Now that would be a plethora of numbers, but we also see that has to be following that the equation equaling zero. And so one important thing to know is that if X and Y are equal to each other, we can play in some values a trial air to see what kind of numbers will get. So say, if we plugged in 54 x and Y, we're gonna get This is really a 25. Same thing with four. You have any 16 3 you end up getting nine. However, we still have this minus one. Over here. It's about me. Is whatever the value of this is, it must equal one. If we moved over this native want to this side, which means no. These numbers 345 work the only number that work would be one. He's been plugged in. One. We're gonna end up getting one equaling that one over here, but we can bring down. It also means that negative one will work because if employees in native one going to get one minus one plus one in the one that we moved over again, appear to the other side, we will do for each other. Here you go the X and Y guys at work or only plus one in minus one. So now that we have four combinations for ex and why we can plug those into this and see what I tell you to get. So now we know these two values are not mylan or maximums. But this too is a maximum. This negative two is a minimum because here, of course, is in between those two numbers. So the maximum occurs at and then the minimum occurs at negative one for dignity. And those are your maximum minimums for the

In order to optimize the objective function F year. Given the constraint function using the ground, multi players will go ahead and first take the partial with respect to acts here of the F function. And that would simply give us one because the derivative of X would just be one. And why would be treated as a constant? Similarly, the partial with respect to why would just be one that derivative switching over to the objective function The partial with respect to X would be two x on the partial, with respect to why would be to why and then we're going to use the three steps the co authors give us here to set up three functions or three systems of equations that will set up. So the 1st 1 is the partial. Um, we're going to set this function equal to this one with a LaGrange multiplier in front of it. So we're going tohave one he calls well, a grand multiplied by two x similarly Well, dio this, uh, partial with respect to why is going to equal Legrand bye bye to your wife? One You cool with their arms died to wife and then lastly or third constraint is that, um we're going to use the G function that were given. So we'll go ahead and write that and balloon here. And then at this point, we want to find a way to solve. We could do this a lot of different ways we could solve for the LaGrange multiplier we could solve for X or for why. So I've chosen to use equations one and two and basically just set them equal through each other, right? And so, in other words, if we moved the two acts over to the other side and then the two y over to the other side, um, we would have one over two X is equal to the what grounds multiplier, which if we then substitute that with the second equation, that would just be equal to two. Why? And then cross multiplying here we would find two X equals two. Why or simply that just X equals Y. And this is what we could then use to substitute. So combining these tubes out, we could replace, um, why, with X in the third equation here, which is what we'll do next. So, using number three and say X squared waas substitute X for y ex word. It calls one or we know that that is going to give us two x squared. They call the one and then last way. Taking the square root back up here. We're going to get that X equals positive or negative square root of 1/2. That's where Rita one would just be one. So it's really room one of one over route to And then same idea for solving for y, and you can see that there's going to be some symmetry here. So now, replacing, um, the ex turn with why in the third equation or the, um, the constraint function that they give us? Sorry, this is going to be cool to one again over here. So combining leads. Two. Why? When is equal to one. And then we're gonna find that why is equal to plus or minus off one over route to. And this is, um, going to tell us about the max in the men. And in this case, the maximum would be the points tangent to the circle constraint off up into the right. And those would both be positive values. So that would be one over positive fruit too, comma. One over positive route to Unless he's both would satisfy, um, these equations here. And then the men is going to be down to the bottom left of that circle where both these values would be positive. Negative. Excuse me. So negative. One over brood too. And negative. One over two. Again. There are different ways to solve and you can try those as well. This is just one of the ways that we can solve easing will grunge multipliers?


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