Calculus 3 Lecture 13.9: Constrained Optimization with LaGrange Multipliers

you have measured I'm going to teach you how to do math magic to at least the starting spot of math magic today it's an amazing concept it's called constrained optimization and we're going to learn about something called the grunt lagron giarla grange lagrange multipliers we've done unconstrained optimization is called absent maximum we did a last section all right 13.8 but we've also done constraint with three variables but if you notice we had to change one of those variables into the other two to do it I'm going to show you how to eliminate that necessity here's the idea all we're going to do is talk about the idea we're going to get down to the at least the equation what I want to know introduce the Lagrangian and that's it so here's the idea take a function notice a two variable function this is a surface and take a constraint where C is a constant listen carefully because I do not want you to miss this recognize two independent variables equal to a dependent variable this is a service right recognize two independent variables equal to a constant this is a level curve to a service do you see it this is a level curve to some function that should make some sense if that's a surface and I let the service equal a constant I get a level curve of the surface transfer okay without me now consider this if I if I set it please watch carefully IDG don't be you don't blink you'll miss it you miss if you blink I start saying this one equal to constants I'm going to get a whole bunch of level curves you guys get it a whole bunch of them so let's let's look at some just made up old level curves level curves of f of X equals c1 c2 c3 did I doesn't matter we're getting some level curves on that surface and let's look at this level curve check it out if this is a surface it's going to have a whole bunch of level moves at different values of C default this is already a level curve the point is there is going to be an overlay of the level curves of this surface that equal to devised so you can develop groups there's going to be contour plot you see all the different level curves are talking about rapport there's a whole bunch of them but that's also a level curve verify something this and all of these are on the same plane that means there's going to be some overlay let's call this the level curve this one there's going to be an overlay these level curves are going to intersect do you get it here's the beautiful thing if we're if we're if we're on a function but we're also given a constraint listen carefully we have to be on the surface you with me we have to be on one of these level curves we also have to be on this specific level curve where these level curves intersect this level curves you guys see that we have to be able we've got to be on both we have a good strength to be on this broad wave function if the villains are on all we're on all these local curves but were bound to this one this is our constraint it constrains our outcomes this level curve is going to intersect these level curves where they do if that's a contour plot notice that these level curves are getting different heights different heights I said it so intensely different heights or kidding different heights this is either falling look look carefully these level curves are either falling away or they're climbing upwards that means wherever this guy intersects and there's bitterness on that intersects them we have different values of our function where it bounces off one of them we will either get a maximum or a minute because either it's drunk and away or it's climbing up just yet this is what we're looking for everything else is going to be higher and lower elbows are in either one it's going to be higher or lower than that point so this right here that right there gives us that kannst where these things in our stress intersect give us constrained maximum or constrain minimum ready for it are you ready for this Duke employee Minds scabs explode it's going to crush your heads ready you're going to love it because it's going to put all of chapter 13 completely together for you promise you get sick with number 1 do you guys get the surface level curves this is a level curve of some surface and the level curves have are going to intersect if we're going to have any solutions you guys get it do you understand the concept that if we bounce off if these little curves are falling or climbing that guy right there is going to be an absolute max or an absolute min for that constraint to get it don't know which one but these are giving us different intersections different outcomes check it out we're looking for the one that we bounce off of now how about this if we bounce off if the level curves bounce off each other if two curves intersect at one and only one point they share a common tangent or at least at least their tangents are scalar multiples I'm not right now us down you're gonna have to listen to it a couple times if you don't catch it there I don't time their tangents are scalar multiples you get it but here's there's more oh I guess there's more there's more if they're if they have the same tangent tell me something other normals same normals do you get that if they bounce off each other they have a common tangent if they bounce off each other they also have a common normal they have a common normal vector or at least the normals are scalar multiples I'm sorry the normals of the level curves are scalar multiples here's the magic there's that you keep on saying that here it is so the normals the normal for this surface is equal to K times its call it a concept right now a change in constant just second K times the normal to this function bear with me love curves of F the level curve listen listen don't wait for a second before you write the down I need your eyes right here the level curves of F correct the one care the one level curve you care about for G this level curve is a level curve of some higher function so the normals for this function are scalar multiple just K times is constant times the normal to this function that's saying that they're level curves have the same normal or scalar multiple the same normal and not put all of your knowledge of all this chapter together what gives you normals to level groups normals of the level curves that means the gradient for F must be a scalar multiple of the gradient has to be here's what we call that constant that scalar multiple has a special name that scanner multiple is called lambda and that right there is what we define as the McGraw multiplier multiplied scalar multiple the lagrangian estimate is the most important part is that this can be extrapolated for three dimensions which means that when you get optimization with three dimensions you don't have to change a variable anymore because gradients work like that gradients give you normals of level curves or level surfaces and since those normals have to be identically scalar multiples we get the gradient or F has to be a standard multiple we call the ruffle type or the gradient of G how many variables are that is fantastic level curves have to intersect we're looking for the one that bounces off if it bounces off has the same tangent say normal so if we go back to the surfaces that created our level curves the gradient gives us the normals to those services the gradient gives us the normal so little curves really gives normal normal to the level curves they have to be scalar multiples we just showed life save 10 jo se-ho therefore we call that scalar multiple lagravis how we're going to start doing constraint optimization I'll show you exactly how it's done next time but do you see where it comes from chef Anza that makes sense sweet okay so welcome back to Lagrange multipliers and constrained optimization I'm going to do a little recap because I know it's been a few days for us so we're going to come back back at it it's going to be a brief recap but here's what we're learning about section thirteen point nine this is just the continuation of that here's what here's our idea our idea is suppose I give you some function verify it's a service like there's a lot of it okay and suppose I give you a constraint well if I give you constraint equal to a constant power constraints are this is a level curve to some other function some other service so so let's check it out surface service is this going to have level curves in fact if I set this equal to a constant I create a lot of constants I create a lot of lunker's so take our function or surface set equal to a constant there's a whole bunch of level curves make sense now this constraint it's already set equal to a constant this is a specific level group so here's how I'm looking forward with our constrained optimization constrained optimization you do have to be on this surface which means you're going to be on these level curves that make sense but you also have to be on this constraint so you have to be on this specific little curve if I were to drop both of them look this this surface has a whole bunch of level curves okay this surface has a whole bunch of level curves this is one specific level curve the constraint that we have to be on you guys see the difference it's whole bunch of level curves this but there's only one that I care about for that service the one equal to the the constantly gives us that constraint so okay these love curves are going to intersect if there's going to be you solutions they're going to sect so the level curve this constraint is going to intersect a whole bunch of these level curves for that service you guys get the idea what we're looking at is for the point where this constraint level curve bounces off one or more of the locus of the function why we care about them bouncing off why we care about them intersected at one point is this if I'm going to map those level curves or this surface onto a plane which is exactly what I've done that's a contour line and what that means is that from here look at you look carefully please from here to here and here and here the function is either falling where the function is climbing does that make sense so if I look it here or here or here this function right here is either higher or lower than this here it's either higher or lower than this so that one point if I think about the contour mountain this is either going to be a high spot or a low spot consider the country remember these things are either falling away or they're client correct so where they intersect these points are all higher or lower than that what that means is that this point where we bounced software one level curve the one we care about it's right intersects a level curve of the other function function where we're trying to maximize or minimize where they bounce off we either get a high point constrain maximum or a low point and constrain a minimum chance get the idea at the contour plot that the contour plot is shaping the syringe it's doing the shape of Surgeons it's either falling away or it's climbing up do I know which one not really I don't the care because well what I care about is it's gonna be one of them we have other things we can do to figure out whether it's maximum and the point is at that one spot where we touch it one spot that creates a high point of a surface like this or a low point of a surface like that does that make sense that's point now we start studying surface yeah but we set equal to constants and we get a lot of cruise surface yeah we set equal to a specific constant etiquette strength that's one specific level current when they bounce off that's what we're looking for constrain max you constraint min now we studied the idea that if a function intersects another function at one point they're said to have a common tangent or at least they're tangents are parallel so so let's let's put this into perspective here please listen carefully the level curves level curve Locher have parallel tangents do you follow that okay say common tangent or parallel tangents however if the level curves have parallel tangents think about this if you have parallel tangent that are like this parallel tangents all your normals also parallel yeah you feel like a mind sort of motion so if your tangents are parallel the level curves also have normals that are parallel level curves superlatives check lopressor fairly normals check that's what we're talking about the normals to these curves at that point is common or at least they're parallel and others with me now normals are vectors tell me something you know about vectors if they're parallel tell something about that scalar multiples so if if our normals listeners we're putting together okay so the equation I gave you last time actually makes sense to you now if the normals are parallel the normals of these level curves must be scalar multiples do you guys get that idea so the normals must be scheduled multiples yeah love burgers vet yep level curve of G yep what I know is that the normals to those level curves at that point at that one point the normals to those level curves have to be scanned and multiple so the normals four level curves if f of X Y is K times let's call it a constant just some scalar a scalar mobile to the normals of the level curves of G of X Y do you guys see then that what we just spoke about is written down and it's kind of a model here if you have we said we said is the level groups of parallel moments but level curves have parallel moments oh there's scalar multiples of each other Chileans feel okay with the idea for real this is the idea of constrained optimization this is the idea of something called the Lagrangian what we do is we we think back a little bit hey everybody what gives us look what gives us normals to level curves what gives us normals to level curves of a surface what gives us that we've been doing for integrating the gradient did you follow this here's a function as a surface if I find the gradient the gradient gives me normals to the level curves to giving there's a function it's a surface but it's a little curve the gradient would give me the normal to this specific level curve but what I really care about I know that those normals have to be scalar multiples the gradient gives me enormous numbers the gradient is you know smokers so we okay so the gradient of F must be a scalar multiple of the gradient of G do you catch on to that say the same thing twice here's the deal we just don't call okay we call it lambda it's called the Lagrangian and it's scaling this is all I'm trying to say it's how constrained optimization works here's what it says here's what says one last time okay check it out so you to bear with me because examples are coming we went we're going to come down about 25 minutes but that's that that's that's it so here we go this gives us normals to level verbs correct the normals to these things I'm looking at this gives us normals to level curves than normal to this this one that I'm looking at makes sense if they touch at one point if they touch at one point they will if they touch at one point their normals have to be scalar multiples so by making our normals scalar multiples we find the point so we're going to find the one point or more points where we have normals that are scalar multiples you guys get at that point our normal scree scalar multiples so make our normal scalar multiples and we defined a point we're kind of working backwards here does it make sense kind of cool all right you know the gradient means normals two level curves so we use that we say we want to find the point where the normals are scalar multiples where they are parallel but have basically the same normal that's where our one level curve constricted intersects another level curve over certain stuff function let's try an example with this I'll show you how it's done I got to tell you right now we're not going to be focusing on word problems just to take a long time I focused on setup and and the operations behind them word problems do happen I'm not gonna be super concerned about it I think that you guys can manage it what I do care about is you're going about doing these problems and not if you're ok with the idea before we go any further okay so how we help me how we do it well the first thing I want you to do is recognize that I'm giving you a service here and they can straight here here's what I want you to do first the whole goal of a pan scope the whole goal is to find land as a horrible lender or to find land so now here's what's kind of nice we already have a function f do you remember you member remember a while back when I gave you like 2x plus 3y equals 6 and I said hey I want you to pretend that's a level curve so you're going to find the function higher that way the gradient gives me the normal to that you remember doing that is we found tangent playing normal lives all that sort of stuff that's what we're going to do so so let's do that real quick I already have a function f but I'm going to create a function G such that that is a level curve it's not hard it's not hard all you got to do is put the function without the constant so what's G of XY going to be everybody please okay listen let's just bear with me here for a second is this one specific level curve of this function yeah namely the one I said it was six that's a level curve that's what we're doing now now come on put it get it what gives us normals two level curves so now that I have a function two variables in a function of two variables all we can do is find the gradient because the gradient is going to give us the normals to that functions level curves and the the gradient is going to give us the normals to that functions level curves including that one the one I care about so let's go ahead and do this right now can you find gradient of F and gradient of G do you remember how to do the gradient it's been a while you take the partial derivative with respect to X I plus the partial derivative spec to Y J that's all you got to do so for F the Delta arts are dealt grading the fact is why why I plus XJ doctor okay with with that one hello for the G do I take the gradient of this one that's already a level curve I want to take the gradient of the higher function that way it gives me the normals to the level curve that I'm looking for so the gradient of this gives us completely to chance relocate with with that it's not hard we spent a whole lot of time so you get to understanding here here's what we've done so far please look carefully here's what we've done we found an equation that will give us the normals to all of the level curves of that surface basically we found the equations candidates and normals to all of these little black circles here make sense we've also found the equation that's going to give us the normal to anything on this level curve itself what we're looking for is the place where they're equaled or at least scalar multiples that's what this says find the scalar multiple and we've solved the problem does that make sense so we go okay well now the gradient of F is equal to lambda Allah grant LaGrant applier times a grade of G so here's the graded investment equal to a scalar multiple the runs will play time's a great engine listen Jen do you see where that came from I don't want to lose you now you learn too much can you do the gradients okay this is easy after that said this gradient equal to lambda time together when Beth that's all I'm asking for you with me now let's use some logic here can you distribute the lambda do you understand that for two vectors to be equal and these are for sure vectors that each component has to be equal so let's use that here's what I'm to do with it just what I know this Y has to be equal to 2 lambda do you see what do you see signal coming up forever do you see do you see why do you see why I felt like Michael Jordan ever second doing math okay can't jump that high okay if you do see it what's X equal to freely all over but I imagine these are three lambdas that we said yeah this quite says fun this coins seriously those do you guys see what what this is so set this x component equal to this X component this Y component equal to this Y component you with me and then write down the constraint ready ready ready ready ready but - your goal in life here's your mission in life in this section in this class it better not be your mission life doesn't suck or what only it's basically my life doom Apple I love it though it's great we do math all the time it's fantastic job but no way that's I need job security you go find some else to do okay so I'm a math teacher here so here's your main goal in life solve and do whatever it takes to solve for lambda and then plug it in that's your main goal in life it can be really really easy like this one look look I have an equation correct can I substitute something in for x and y that lets me solve for lambda this is the easiest type where you have a direct substitution we get okay here's two times three lambda plus three times two lambda equals six so take Y god take X got it make your substitution solve for lambda your main goal in life right now in this section is solving for that that LeBrons multiplier it's basically like what you do on like direct variation inverse variation you have that constant you got to find the constant first that's what we're doing so here we have well how much 12 lambda how much is lambeth have perfect perfect now wait a second do you have something up here that would map lambda back to values of x and y yeah these two we use in concert all three of them so why it was to lambda x equals three lambda here's what it tells us remember how we found critical points and all that stuff and then we had to plug in and values all that stuff this right here does everything for you because we base it on a contour plot I know that this right here how many critical ways you have one provable tell me specifically what it is perhaps one that right there is the one group of what we have that right there is the one point that this constraint is it's a little group that that level curve that constraint is going to bounce off and intersect at one point one of these level curves the one where lambda happens to be one-half we plug it in we find that one specific point here's the the last piece of information that I need to know all constrained maxima and minima will be found at those points so I know that what these points give us are either maximum or minimum of this function in this case then did you catch that you might want to write that one down no not constraint these are absolute with the constraint so maxima and minima will be found at solutions to this maximum minimum will be found at solutions to this did you write the down by a down maxima and minima will be found at solutions to this to find the actual match riman you take this value just like anything else plug it in so what is the what is the value when we plug in this well that's pretty easy one three-halves times one is three halves now what is it I said maxima and minima happen at that point but is that a maximum or a minimum that takes a little bit more work sometimes if you have two points it's easier because you can tell ones the highest ones the lowest you can tell that if you have only one point here's what's happening this is either the bottom a minimum and everything's climbing higher or the stopping of these climbing lowered this particular one is a max it's a maximum how I can tell that I would have to do the second derivative test that gives a saddle point and based on the constraint what's happening years for this function the saddle point gives this basically some hyperbolas here where they're going one way along the x-direction below and one way along the y-direction above and where that plane intersects that you see that that's a that's a plane going upward where that plane intersects those it's in one region that everything is climbing higher those that saddle points is climbing higher you need a second derivative test to tell it but that right there is a maximum value I think I'm not going to going to right now for the same time but that's a max I want to show you two other ones one other one what we want to do them skip over we have and easy for this was the easy example we have a really hard one once I'm going to on hold for next time because it takes a little while to explain it and then I'm going to show you how to do this for three variables so I'm going to do the three variable version first it's not any harder than this that you want to show you can do it that's one of the beauties about the Lagrange's multiplier they could train optimization is it this stuff it doesn't matter that you can eliminate a variable it works with all of this because gradients are easy to find in three variables so I'm going to show you that one and then we're going to call it good for now and I'll come back and do one more of these examples next time to really fill it out okay so we're going to try to optimize this function with this constraint verify this is a mods like a surface of 4d yeah it's crazy this is a constraint that three variables equal to a constant this is a level surface of some higher function so what I want to do right now here's the process keep this function the way it is make a function for this such that this is a level surface of some higher function let's do that now all G can you tell me please really quickly what that function is going to be come on tell me what about the one do any of the one verify that this is a level surface of that he's jump okay cool now the next thing whether you get it or not I want you to get that's why I spend hours talking about this stuff whether you get it or not the next parts actually pretty easy find the gradient for this find the gradient for this because that is only matters to level surfaces the case and they have to be scalable so I'm gonna give about thirty Seconds to find me the gradients got an X Y z IJ K climb to the creative XYZ ijk set them equal by a scalar multiple call lambda or lacrosse multiplier let's double check our our gradients here did you guys get say ingredients that I got ya now yes yes no no okay so if you want to do a shortcut right now and just go x coordinate lambda X Co X component that that's fine you really need to build it out of it well you know what I know that the gradient is equal to lambda times the other gradient that's fine too so we have I plus 2 J minus 2 K is equal to lambda times 2 X I plus 4 y j + 8 ck if you want to build out like that that's absolutely fine the key point here is that check your x components your Y components in your Z components and get your series of equations you need to do that so I want to go really quickly come on I know you're losing focus here let me have your focus can you tell me what the first equation is going to be please oh I didn't hear a word you said on is it - excelente - okay fantastic someone else come on quickly what's the next one perfect and lastly - yeah fantastic now the last one I want I want you to write out the constraint whatever the constraint is that's your last equation that's an important one you get assessed by that hey how about this do you guys see a way to take these three equations and use them in this one do you guys see a way to do that how would you do it come on what would you do so for what if I solve for lambda I'm not going to have any place to plug it in for free if I solve for x and y I like here I have some places to plug in and let's let's try that can you double take my work I get 1 over to lambda 1 over to lambda and 1 over negative 1/4 lament you guys get the same thing I got hey now that we have these variables and a place to plug them in do that take these variables substitute what they're equal to into that equation what you're doing here is you're finding the 1 value of lambda that's going to create a level curve that has well lambda that creates the same normal for two different level groups and then where that point exists that's it max room it then we could simplify it so if we simplify this what I got was one or four lands squared plus one over two lambda squared plus another one over four lambda squared and now if you okay with with that so far for real substitution put it in there hey how would you solve that come on quick we get them quick denominators okay sure common ground or whatever multiply everything by the LCD doesn't matter if we multiply everything by 4 lambda squared then we give 1 plus 2 plus 1 equals 4 lambda squared well that means that that's I'd ask for lambda squared equals 1 everybody please how much is lambda equal to what is it good we get to hey hey do we have something that mounts us back from lambdas to XY and Z values so the whole idea is creates yeah set them sort of equal the garage multiplier says they're scalar multiples that's what it is set component for component equal try any way you can any way you can to solve for lambda then reverse substitute so once we find our lambda we can plug in each one and negative one to all three of these components and figure out what that point is do it go ahead try it go get the same things that that I got now here's what we said before what we said was at that point verify that's one single point XYZ one single point for that function head not just note it also satisfies this thing that was the whole key here so we have one point we have two points one of these is going to be the max one of these is going to be the man maximum minimum are found at solutions to this that's that that was the statement I made earlier so what we're finding are maxima and minima what if the highest value is that's our max where the minimum value is that's our men how do you figure out which one's the biggest one which one's a small assignment plug it in to where funky original right there plug it in right there so if we plug it into that f of this half of that plug them in I've already done for you two and negative two if you can double-check me I think I think I got those ones right hey that's kind of cool right there's no checking there's no guesswork what's the maximum value what's the maximum value this function based on the fact that we have to satisfy that can strength what's the maximum value yeah that's called a constraint maximum it's the highest that's that surface gets but based on the fact that we gotta satisfy that that's kind of cool right so this is the max and this is the min that's the idea these are the two values we have those uh those scalar multiple normals where we create that point that we bounce off one level curve bounces off another one we happen twice one of them is the high point one of those over point pretty neat huh so fans of the other standard tell you what we have one more really hard problem it's hard only because of the substitution it's obviously great it's pretty easy right greens would easy this is pretty easy but doing this the systems can be kind of tough so we're going to try that next time so as I said a hard problem hard problem we did one really easy we did one three variables now we go on really hard the thing that's going to make it hard is to algebra algebra behind it if you recall we're working on constrained optimization those lagron should look the calls multipliers and the idea was take level curves find normal slope curves its gradient they have to be scalar multiples which ever going to share a common note that about this plan so when we're given these two functions okay here's your surface here's your level curve we must create a surface for that to be a level curve of that's basically just take your variables good pets that's the surface and you can verify it if I set this equal to 8 I get that specific level curve that is on that's percent on the Jew is no metal after that we find the gradient of both of these surfaces this is 2x I plus 2y j this is 2x plus y I plus what two or three yeah we do that way J those are the two gradients those are the two normals to those level curves so those services are you follow me which means it's a normal to that specific level curve and we're saying that we want to find the points where these things are scalar multiples that's why we said the mclamb does some constant that we don't know but we do know at the points we're looking forward about that the level curves bounce off each other they will have normals that are scalar multiples so we force it to happen but finding the lambda the scalable well it makes it happen okay well that means that 2x plus y I plus 2y plus XJ equals lambda some skid these must be scalar multiples for us to have a common tangent common mobile or force those local curves bounce opportunity that's the main part that's actually the easy part I'm your show manager see where all the subs coming from now the idea is if I have one effective air by vector equal to a scalar multiple of another vector the components of those vectors I and J have to be equal so I know that 2x plus y must equal lambda times 2x does that make sense to you I also know that 2y plus X has people lambda times 2y those things have to be there what I told you what I've told you pretty much this whole time is write these down write down your constraint the whole goal in life for you in this section is to solve Lando I've given you two ways and there are two different ways to do it I'm going to give you this third one it's a third way to do it I've given you three completely different examples to show you three different techniques this one's not going to be that easy there's a lot of algebra behind it this was these apart do you understand that the idea behind this that's pretty cool right where we're setting gradients equal to a so they're by scalar multiple top of the garage multiplier that's pretty much it our goal is to find that and then substitute back in to solve this thing so we're going to do that we're going to we're going to go very quickly it's all algebra the algebra and calculus as you know is the hard part so I'm going to show you how to do it I'm going to go fast though I'm not going to ask for a lot of comments or feedback I'm just going to show you the technique here so the technique this one I would solve for line and I'm going to factor out two eggs so we get y equals 2x lambda minus 1 this is on the solve for x if we do that we get this solve for X subtract the two y factor in the same version you guys I'm Shanna valves roam at one now now I have to get rid of one of these variables it doesn't matter which but I'm going to take this and set it in front of Y here so x equals 2 times 2x lambda minus 1 times lambda minus 1 here's the 2 here's the Y here's the lambda minus 1 can you follow that ok so what we're going to do since we have one variable now I'm going to distribute I'm going to get 0 on one side and I'm gonna use the zero product property to figure out some values here so let's let's go ahead and and it's this at least it should be this so I have for X lambda puts lambda minus 1 squared I move everything to one side we're going to have this 4x lambda minus 1 squared minus X equals 0 and that's the key that's what I want if i factor out the X i factor out the X we're left with that I listen I do want to move fast I don't wanna leave you behind algebraically hi you guys okay I'm it's a horrible substitution but it's kind of basic substitution solving for wise subsidy and after you solve for X you guys get that that's the hey that's a system you can do it our goal is to solve lamda that's what I want now I still have some X's in here so I'm going to get everything on one side of it a distributor and everything on one side so 4x lambda minus 1 squared minus X unaffected the X away now because that's a product and this is a product of two things equal to zero I know we have two cases x equals 0 or 4 lambda minus 1 squared minus 1 equals 0 that's got to be the case one of those two things has to be the case for us to get this equal to 0 does that make sense now here's the issue I want to backtrack with this one so you understand that this is irrelevant here's why if I let X equal 0 x equals 0 then 0 and 0 means Y would also equal 0 do you see that that x equals 0 and y equals 0 is that on our constraint now because 0 plus 0 does not equal 8 so x equals 0 does not lead anywhere for us it's an invalid solution because it's not on a constraint plug in x equals 0 y is also 0 you go on no this one is no good this guy I've told you from the beginning sulfur lamda you have to solve for many if we do we have lambda minus 1 squared equals positive 1 before that we take a square root we get lambda minus 1 equals plus or minus 1/2 if we take this and separate it lambda minus 1 equals 1/2 and lambda minus 1 equals negative 1/2 if we add 1 we get lambda equals three-halves and we get lambda equals 1/2 you guys okay with that okay now now since we have something that gives us X is something that relates X as two y's we're going to take each of these and plug them in for lambda it's going to give us an equation so let's use let's use this one y equals 2x 3x minus 1 that's 1/2 y equals x is where to define what that does that gives you a substitution to plug into this constraint so we did all this work to find a substitution to plug into that constraint so we go fat fine limit great plug it in what it tells you is not an actual value of x it tells you a substitution for X or Y it says yeah plug it plug it in here that's exactly we did is it's right here this right there plug it in it's telling you why it was X now we take that and we said what ever you want let's choose x equals y or y equals XL care I'm going to choose y equals x then x squared plus x squared equals 8 - x squared equals 8 x equals plus or minus 2 if x equals 2 then y equals 2 if x equals negative 2 then y equals negative 2 those right there are two points that will be either a maximum or minimum that's that's the plan which time we got four minutes left okay very little if I do the same thing here we got Y is 2x and we're going to have 1/2 minus 1 all right now that means that y equals 2x times negative 1/2 well that means that y equals negative x however if I take that and plug it in x squared plus y squared equals 8 when I square a negative I get I get a possible so it's still going to x squared plus y squared equals 8 it's giving us our expert as I screams like it's giving you the same things x equals plus 2 or minus 2 however if you go back to what we use for that if x equals positive 2 y is the opposite sign and if x equals negative 2 then y is the obvious of something that's what that says so we get some points that look awfully similar we get the point 2 2 negative 2 and a 2 2 2 negative 2 negative 2 2 so out of those 4 points down let's write up here out of those 4 points you tell me this and then we'll we'll be done out of those four points how do you check them where do you put them to see which one's the biggest and which one's the smallest already doing you put it right there to see which is the biggest small so we're going to evaluate F of all those things here's what the you try on your own this one is 12 this one's also 12 they took the square this one's for this is also for hey what's the maximum value we attain out of this constraint optimization problem we learned earlier that these have to create maximum or minimum that's what they do all maximum and will occur at these values of with Orlando what's our max value where's it occurred two different points what's our minimum value where is it occur two good points that's our max constraint maximum that's our min it's very minimal I know the algae was fast you might want to go through the couple times on your own but that's the idea

Channel: Professor Leonard

Views: 213,793

Rating: 4.9316769 out of 5

Keywords: Math, Calculus, Calculus 3, Leonard, Professor Leonard, Mulitvariable, Function, Optimization, Constrained, Gradient, LaGrange, Multiplier, LaGrange Multiplier, Constrained Optimization

Id: nUfYR5FBGZc

Length: 58min 33sec (3513 seconds)

Published: Fri Apr 15 2016

Reddit Comments

[u/Mistapurple]

I believe Khan Academy also has a good explanation of this concept as well

[u/OnePunchFan8]

btw does anyone happen to have the solutions for the math 2130 midterms of winter 2017 and fall 2018?