Math for Economists - Lecture 12

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Math for Economists - Lecture 12
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Statistical hypothesis testing Definite quadratic form Point (geometry) Multiplication sign Insertion loss Order of magnitude Derivation (linguistics) Mathematics Kritischer Punkt <Mathematik> Many-sorted logic Positional notation Lecture/Conference Natural number Matrix (mathematics) Negative number Square number Theorem Determinant Position operator Social class Process (computing) Eigenvalues and eigenvectors Computability Maxima and minima Linear algebra Calculus Numerical analysis Connected space Vector space Triangle Right angle Family
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I'm gonna repeat a little bit of what I did last time to sort out on the border can refer to it as the 1st thing is dealt with a positive and death positive and negative definite matrices 1 through this whole thing again but I want point something out of these 2 possibilities for a matrix square matrix and we we learned that a test for those things you can check that the test that I gave you but I want point out of their minimum book a couple of times Hiram 10 . 1 5 another waited tested positive and negative definite is rejecting the idea valleys and it actually reveals a load of y As these words are you so Positive definite matrix this theorem says that all hiking values are positive And if you remember the negative definite definition had this weirdness to where where the miners were switching as I took those determinants 1st always negative the sector was positive the 3rd 1 when we look at the height of values you see why it's really a negative definite nature In this year of a new book 17 all nite in value I don't think there's is a connection between the 1st portion of the class the 2nd were doing calculus but these Linear algebra concepts creep in In this portion of the math class if we just think of it that way sometimes you gages glance at the Matrix in all immediately raise because For instance if you have a matrix like this For even if I have the upper triangle or something The Values if you if you go on computers and I couldn't go back and do this right now but if you were to just for practice with family values you'd find that their 2 3 and 5 so when I get back in his glance at that you know that it's positive definite by knowing that these are the ACGIH values and their positive and therefore the nature is positive When you can't see the agonizing Reddaway your news via the Internet the last time that did this f of x equals X. square him and we analyze it likely would calculus class don't do that again just a point out a few so if I get this and I wanna find out what the maximum in are you know the answer but You go through the process to take the derivative you said Betty quotas 0 that tells me that x equals 0 and so that's a critical point and then we take the 2nd derivative accuse me to and then because the 2nd derivative positive the man means that at that critical point I must have a minimum an attack that Mexico 0 But it's at the point 0 0 Higher so this To here This is This is like a high value a week he can't really see it here But when I do the other examples where I had ever X Y equals X squared plus Y squared they do this example then we get the 1st derivative the 1st derivative in this context as the and XQY unsteady 0 just like it said the 1st derivative equals 0 here I have weekly agents she left I have 2 x equals 0 his and go 0 and 2 equals 0 and his new wife equals 0 and then when I go get them passion switch That This notation switch to us with the book has so what's this guy this is the 2nd derivative matrix the 2nd derivatives puts them all over here that access to and X Y 0 why 0 1 His to lose robber 2nd derivatives stemming from the past when I put him in the matrix I get that and you can see those of agonized Another thing to communicate about this number here usually do all we care about is whether it's positive or negative or 0 so want to note positive that we can say it's a minimum but the number itself Just like these Agon values they they related sort of the magnitude of the concavity so if I have a half-year will have less can't have It won't be as extremely if I had a five-year be much more extreme so the number the magnitude of the number gives you the magnitude of the cavity in years values Here they're both positive so far over here when we get this positive value here you can kind of think of there being a vector and eigenvectors pointing out that indicating that the user on opening upward and these guys here these Agon values remember those come equipped with eigenvectors and so the same kind of concept is going on here I've got to lagging sectors in their pointing out that makes it a concave up so the other possible I drew on and like they're not the the same vector but they are But because there 2 over so the other possibility is that both O'Meara pointing down that would be a situation where I had negative to the negative 2 and then there's a 3rd possibility which doesn't really happen here and that is that I could have 1 of employing up the moral pointing down will get to that That's that's just gonna make a connection between all the eigenvectors and high values and can't cavity and everything really is related that's why in this course we do those 2 subjects because there is an intimate relationship between pocket so now what I wanna do as wanna take This year of the money finishes so we have this guy here is positive definite so just like if this was positive that means that we had a minimum so we have a minimum . 0 0 0 that's the point here is So let's get a recipe before we start doing so real-life problems is just kind of a warm-up others get a recipe for how you do all these things
I If I wanna encompass all the this scenario at the same time I can't really specify that we only have 2 variables for a lot of the problems were gonna do there's only 2 variables maybe 3 but that when I write down the general theory I just wanna talk about if I had and variables so let's have a function goes from our until all of this also encompasses that situation that that's a Jewish you will be were and to is said to variables and his red that downstream and means that many variables so the 1st step is to get the 1st derivative but in this context the 1st derivative has lots of 1st derivatives depending on how many variables you have sold just convinced that all the saying I find the gradient that the the 1st Just Like a single variable go get the 1st year Then what we do it that would regulate I said illegal 0 And this gets you what this gets you critical points Bellamy address Vocabulary here And that the book you guys using the column stationary points that's fine that's their code in economics setting the reason you use the term stationary point because it actually it alludes to the fact that when you solve these things you get equilibrium so equilibrium if things remove around when you finally get an equilibrium point that's a stationary point or another way to look at it different if we're looking at this paraboloid here if you would drop a ball in here it would go like this for a while but eventually it would stop at the bottom and that would be staged as physical reasons why you might call it a stationary point but also in economics at an equilibrium Or a maximum point at at the top of the hill pockets of so that I a keep column critical points is Lester right for me and just use That term but those are interchangeable So the wants I have The critical points that I'm gonna take those and plug cancers there might be more than 1 just like with in regular calculus when you had as the derivatives said Eagle Azeri make it 5 different values The derivative being 0 you make it several years so I say answers at plug those into The hatch The 2nd derivative matrix just like we plug-in The critical point in the 2nd derivative here you don't see it because it is no X value here but if there was a plug-in next facility Concavity woes OK so applaud those into the Hessian and then we could conclude once we do that is we have a Max if the Hessian positive cursory negative definite and we have amended that is positive definite awaited think of that It is if If I tell you that something's positive definite then you just wanna think of concave are so just like positive to over their meant concave up positive definite mint can't give up so you when you're sometimes you know I get on the test some was positive definite Minneota Max does the opposite sex so positive definite means its opening up like this only a minimum negative definite is a maximum Blucher recipe a hot what what I haven't included here and a recipe is if the matrix if you get the Hessian isn't either positive or negative Avenue included that wall at that a little bit later that's going to be like the inflection point we have enough to do that now silicious focus on
So do a real-life example there real master like example Based on that 2 variables and that this combination alone in all kinds of combinations to unilaterally really in a deal that see what it looks like the Masuda of Wolfram Alpha or something like that but we can still do all the analysis of the recipes is 1st things 1st go get the and what's would uranium that means I take derivative with respect to the 1st and the 2nd variables day with respect at 1 negative toward their that 0 This is 2 x 1 that's 0 In the the derivative here with respect to x 1 is next Dallas due derivative with respect x 2 to hear this 0 this 0 minus 2 x 2 And it is safe to do so said equal 0 so the their requires the solace system of equations and that's why we had to practice that for the 1st portion of the class because it comes up for ordering the calculus source said that equals 0 when I say the gradient is equal to 0 the gradient deserve that Here got multiple components my 0 vector hats What components that is this parts equal to 0 and this article 0 that is a system of equation If you want you can go and do this with the linear algebra that we learned that 2 variables so how about we just go directly so let's take Other take top winery here that says that for minus Q X 1 next to equal 0 analyst X to hear and X 2 As equal to 2 x 1 minus 4 And then scary Pierce says to minus 2 x 2 us X 1 0 but I've sulfur X 2 here so I can take this substituted back in and basically way doing Now some incorporating 1 equation into the other nite got both equations in in 1 equation I do that when I saw solve have the answer to Time to X 1 minute for plus 6 1 0 2 . 4 x 1 minus 8 1 0 0 put together Negative 3 x 1 9 6 equals 0 What plus a yes thank you knew that 6 was I think you plus a negative tutors native forest plus yet please help me if find summers was go back because I it can you So now X 1 thirds and then X 2 is equal to 2 times tempered minus 4 witches But with the 23rd minus 12 pocket so we got a critical point at 10 there is 3rd base or instead To there we found the critical point there's only 1 now let's go get the 2nd derivative matrix that's not too that the derivative with respect to x 1 is negative to And then the derivative with respect to x 2 1 because of young Sarena's won't be the same as a derivative of that 2 with respect that want me very right out Were all new it this However 1 1 1 2 2 1 2 2 days so at 1 1 already over here that's equal to a negative to death of a hack 1 is equal to 1 and at 2 to the derivative of 2 with respect That X to negative so there's my matrix negative 2 1 to make it as glance at this maybe you can bet that kid to see exactly what the eigenvalues are severe would do the regular analysis of that Some positive definite pesos When you doing positive definite just check the upper left-hand corner At this Point you only have 1 candidate it could only be negative definite because they were positive that this would be barred right so we're only in the negative definite case and then what we need for negative definitely need this this determined so the determinant of a 1 that subordinated to And the chairman of key to his goal think that's equal before minus 1 is 3 so it alternated it went from negative to positive so this is negative dead and so if we have a negative definite that have Max think negative definite Pentagon 1st example to practice on
Search to get a little more complicated any questions about that yes it would be Final Yes OK OK so there's 2 questions is that what you find the critical point indeterminate it's a maximum and then they can the other question is what is the actual maximum value of some of the functions of 1st base here so the Mac's value that is what we plug enough points so it's
It's At the end of the 3rd and actually calculated that it is aged 84 so yeah you could you could say we have a match 10 3rd Parents before they avoided little bit since ask avoided it is because of the fact that gathers to separate questions it's fine the critical point indeterminate that's a maximum generally care with the function values But then I could say what is the maximum value of the function and that they answer to that would be be 84
The war There's your function assumes got a little extra duty year it's got Q terms of skin and make a little bit more complex result J. let's get started the recipe says
We're gonna go get the gradients that 1 of 2 1 is 8 x 1 9 x 2 plus 0 minus 1 square I said he a 0 It isn't Add to that 0 the derivative here is negative the derivative is thought to have 2 0 This is not a system of linear equations remembered looser Opel class with that kind of vocabulary this would be As soon as you have an X squared term that is not a linear equations you can even really go back in use that matrix method that we used before here you can have yet to rely on substitution but it's not too bad because we have this equation right here I can immediately solve for 1 of the variables in terms of his or things that happens as we get more complicated systems of equations when you doing the calculus is not always get linear systems so that scary here says negative X 1 plus 2 extra was 0
4 Here that means X when 2 x 2 And then I can take this year and plug it in Florida X 1 year and the 2nd version here 2nd ahead 8 x 1 minus X Tulu minus 3 eggs 1 squared equals 0 And I think there's a put it in Virex 1 and I'll be able the sulfur to had 8 times 2 x 2 that's the first one minus X to minus 3 times but that in have 2 x 2 square Equal 0 It's 0 6 teams next to minus Next to minus 3 Be careful here is a square surrogate for x 2 squared times native 3 so that Israel that Because of the final thing we get is 15 Next to minus 12 that students squared equals If you believe that
I The all with you so far now after rely on being able to solve this is this Equation rain gear system anymore single equation but we have really done anything like that but that's not to say that
X 2 and both location so factor out next time we had 15 Minus 12 2 0 so I get to values for acts to their next 0 Or is it seemed 12 x 2 0 World 2 equals teen We are for 5 Focus so I've got to values the next to but then for each 1 of will get a value that's 1 x 1 is equal to 2 x 2 x that implies for this 1 here Too X 0 0 implies that 1 0 thanks to peoples 5 force that flies X 1 2 times 5 force which is a I had to critical ahead 0 0 and they have 5 have about 4 stupid excellent offers itself so now at this stage trade here I just said degrading equal to 0 and I found the critical point The panel operate there Let's go get the hatch the Hessian the 1st derivative with respect to x 1 is going be More things in it it'll be 8 minus 6 at 1 8 9 6 x 1 that's the first one here and then the derivative of the 1st line there with respect to x 2 his negatives 6 using negative 1 just respect in the 2nd line derivative with respect X 1 his created 1 derivative with respect to 1 place here where The plug these points in determine whether I have a positive or negative debt of high-value which added 0 0 Then I put 0 8 Negative 1 negative too And so is that positive definite or negative definite it only has 1 possibility can only be positive definite just because the upper left-hand corner now you checked the whole determine its 16 minus 1 which is 15 persist positive so positive definite you would think that is concave up so we have a minimum here so it's at that location 0 0 if you wanna find the minimum value the function it's also have all that information in that warn that happens at 0 0 but the function value is 0 OK something different Skinner happened with the next 1 that's been leader center
Our next topped sometimes you get critical points of the matrix that you get is neither Positive nor negative definite that we have what to do and in that
Check that Checking it now This Point here 5 halves quarters Wanna plug-in up 5 halves here Negative said they wouldn't fire has 6 6 over to series so this is a minus 15 is native said and the rest of us so here we only have hoped for negative definite just because the upper left corner is negative so I'm getting again native definite than whole matrix that determined that as positive but it's not so That out of the way you want that the negative 7 and 8 to that The whole thing that's negative 14 minus 1 witches So this is a very positive definite nor negative death is is not a Maxine's not reviews he said What is going to turn out to to be this is saddled with that's like an inflection point but take a look at that little If cross our other back
But It's a saddle point this is a little bit like an inflection point but when you're in this situation you have so many other things happen so as not a direct analogy inflection point be a little bit from that source go back to match to a look at it as much as I can use my standard example of why people X squared inflection point you have 1 so to go a Wykle's execute So here's the thing about the inflection point verses versus a minimum of the minimum of just compare or maximum If you're When you have a minimum if you're on a scale of 0 walking around on the grass if I walk Towards this point I feel like I arrived and that no matter how I arrive festered over a year I walked Point a server here I feel like arrive at a minimum so there is no ambiguity it's definitely a minimum here if I was walking on this graph and I was walking this way I would feel relieved to have gotten top feels like they had a maximum and if I'm coming down like this I feel like they a minimum because it it tapers off I know it's not a maximum but it feels like 1 because the Cannes cavities which it kind of comes to a Tianjin where the horrors horizontal Tianjin so you feel like you've arrived at the top figure to walking that way Where is he going this way you feel like you about so of got both Aspects to it in math For much more of a saddle point is Some like this is group That's a saddle point is the actual saddle point here If you look at the mountain range around this we weird Saddleback mountain looks like a saddle But here's here's the deal with the saddle point is that it's like a mountain pass a mountain passes is a saddle point because so think visas to mount and if you wanted a crossover mountain range you take point to get over erect as you would wanna go over this pointer this point this is the lowest point but as you arrive you've reached a maximum so you have both Macs in a minute the same time So here's see that if I'm walking The move up the mountain this way and I feel like I had a maximum and so relieved to get to the top but if I were walking this way And I arrived at the same point as feel like minimum so Actually we all have a saddle point you come equipped with a saddle point all of us he isn't unavoidable so deflected and and is crawling up your legs It's a maximum feels like it's a maximum would gets the top but I'm looking at the profile and can't walk Stanley chest feels like it's a minimum so that's why I So comfortable on the saddle right we take our saddle point do put it On the saddle point that feels really look at So This because we have a maximum and not happening simultaneously so when you build the seat you build the maximum in there at the same time so that's really that's why we call a saddle point the mathematical equivalent of things that obviously horse back prayers discovered a long time comfortable seat has a maximum and minimum on the sea so now we have to do a little bit of a math however we determined that we have a saddle point without drawing a picture in this particular example we have both we have a minimum and a saddle point so you can imagine the services come a complicated over here get this little minimum that we found them they goes often over here there's a saddle point to you can really see these things but you figure out what's going on
That no So let's have a splintered example of this is a saddle it's just like I did this form before you before but may not remember it's just just like eggs
1 squared plus XT skirts with slight variation is put a minus their causes the graph to be completely different results tour now says it won't take long Yeah ingredients need to derivatives to 1st drew saved to X 1 2 minus 2 x to to and said that 0 said that equal to 0 that forces X 1 0 and that few people 0 0 at a critical point 0 0 Then let's check that passion the derivative with respect x 1 hears too Derivative with respect to x 2 0 derivative with respect X 1 0 derivative with respect to x 2 As negative so you see the eigenvalue positive ID in value in a negative light in value that that represents that at 1 point we've got the Cannes cavity being pulled down 1 direction and being pulled up another direction so if you think about the original example on their shows you can't cavity was to here this Agon value being positive to and this being made it 2 got 1 going in each direction and that's that creates that's what want codified won a recipe Typically what's done and even in a duties that would have been a course that here that focuses on this matter Calculus of several variables they they give you a little bit more as far as far as analysis of give that you know your book doesn't have that All I can tell you talk about the positive in Death The positive definiteness of this this is me there but the way you Telesat were about to tell you that I want to tell you that example that the determinant here was less than 0 equal to 92 Which is less 0 and if you think about both the positive and the negative definite case both of those determinants are part of parts safe In the positive definite case you had at the upper left corner positive and the whole determinant positive both had positive so the determinant was positive and that kicks in the negative definite case be the upper left-hand corner negative football determinant positive so in both cases the determinants positive so the only case that was left out was 1 of tournament's negative and that's when you have a negative determinant of the Hessian that can hear so straight down every passion universe Alderman universe where they look like they look like this that haven't specified any function I got already down all the universe there
Who the bat A critical point we have a Max if as the Hessian It is negative definite answers can just do a little help with the memorization of Amax at its negative definite can you have a man that that innocence Positive definite recovered all that and so what is it positive definite look here So Positive Yet Order deposit occurs focus so if I just look at that meters what makes it positive definite Icahn taking condensed into a little bit of smaller space I have to have this thing here positive right so that 1 Major So it In other words it has to be concave up in the next 1 direction and then And then also have to have the determinant Of that has to greater it does what it comes down to They have the 2nd derivative with respect to this variable positive and then the whole determined that the negative definite it's the opposite this part the opposite this guy has to be easier and less than 0 start with otherwise you can even in the case but then the 2nd thing is this guy also has to be positive determinant of the in response he had a glance at it would mean that same thing New year and is putting it in terms of new notation but then the other thing is is that time now I can add and would have a saddle point when that determined lessons to see this doesn't fit in either those cases that I was saying before it's positive definite the determinants positive if it's negative The determinants positive other case when the Determinant is negative and that's the saddle and that We have here The determinant there's negative so don't have yet here's the other example here
See this determinant of negative about the saddle point of focus and I know everything needed no stars the Hessian is positive definite A minimum holidays maximum benefit the terminal here is negative and you have a subtle point covers all that stuff And intuition behind that just think in terms the agony of the saddle point if you have a negative determinant attending you had 1 nite in value that was positive and 1 of his negative but to make member the determinant is the product of the idea that so if they determine negative and is the product of Thierry values is negative which is 1 was positive and 1 was so that the finest calls it in 2 directions and you get this settled
That was so fun let's do this with 3 variables the good news is if you get over being intimidated by the fact that we have 3 variables these problems are actually pretty easy because if we were to make a moderate is so complicated
Could make a mess if we wanted to but if we want make easy Then they end being really is so that's a good thing if you just get over that that this 3 variables they turn out to be pretty simple Follow a recipe By just just to be clear we were talking about a saddle point we're only talking about the 2 variables situation when you get to 3 variables all kinds of things can happen and they're not called saddle point yet Last March should be 0 The but because terminal as see that's the only thing I need to know that determinant that's so yet you can have and in that case yet if you had an upper left-hand corner that was 0 you would you could Nevada Maxtor Used couldn't you could been in those cases and then if you look back at the matrix of that should the terms of a scooter through our processor processes 1st get the radiant but in this case I've got 3 deterred Determined to get their derivatives to get back in a derivative with respect X 1 bank is 8 x 1 of these 0 . 4 pretty easy to do that 1 That 2 that's 0 this is forecast to and then there is no other acts to turn over all 0 and then have 3 that 0 that's 0 2 That is 3 and then plus 5 And will agree that the people a 0 Susan what I mean about these these being Kenny easy but just because we can make them too hard because at this stage this is where we get really are messy messy algebra trying to solve a system of 3 variables that you see this right away If that's equal 0 that implies that 8 x 1 minus 1 equals 0 itself attacks 1 is equal to a But that X 1 ran away and the 2nd line That says 4 x 2 equal 0 that immediately tells me that X 2 equals 0 in the 3rd line That says that 2 x 3 5 0 0 that's as the next 3 feet the negative side See that solves the whole system for me is that the only solution so we have a critical point right away At 1 8 0 0 Negative 5 adds And the We have to do it to test it that this is get this guy going with his right out all derivatives 1st just to be clear there so this is degrees F 1 1 and then had 1 too That 1 3 2 1 2 2 2 3 3 1 0 3 3 So Amadou is unjust and focus on this top-line menu all derivatives That The derivative with respect X 1 that a man with next year 0 x 3 That's also is that takes care of all outline the remember when you doing these derivatives we have to 1 equals 1 to race and 1 3 equals 3 wants weaken and other things in this matrix RED these have to be the same as those that do the same path 2 2 on the 2nd line near the derivative of this with respect excuse for derivative with respect X 3 0 and then a fabulous 0 0 these 2 And then had 3 3 3 derivative of this with respect Dexter To get nice diagonal matrix and I can say immediately that this is positive definite because all these are the eigenvalue The idea So Agon values Each too there are greater than 0 that Diaz's as positive Definite proof and that implies that we have a positive definite its concave up that's on That's at that point sources put it here For that point If you want to find out the actual minimum of the function you could plug those points back in but you do that We're back to the Messier ones related to variable bonds because their regular might get a system of equations that has Neck squared term or something But these 3 variable once I promise to keep on relatively simple that any question about that that's basic math to a extension enough for a defined Max isn't it
The back of At the next level of this is But we remain and this was kind of a global analysis find the critical point say whether it's Maxtor and now we're going to do in a restrict the amount of points that we look at it and then find out what the Maxton in his little patch of humanity Gillette talking about us over the energy issue is this day I've got a function of 2 variables and before what we're doing is ok just take the view that a derivative analysis and find the critical point and maybe we have little maximum there may be a L minimum there we could discover them now what I wanna do is I wanna say always buy only allow myself to pick points from this little regenerate here then what's the next
What I'm only allowed the depict points from here if I start plugging in points will I get up here What should I visualize appear ushered should visualized may be down panel wobbly napkin it's flat here but wants plug-in all those points and make it a little floating that there and this might be the maximum rate here and it have meant there and this is what I wanted discover them I have 1 in the middle here might have a little Max there something of this is that this is new scenario In order to do this I think it's prudent if we go back to a single variable and then look at an example what we did this amount to a typical example would be half of that will swear My standard example but OK so will be the equivalent of that Okolona that would be I restrict access to be Between native 1 2 and then asking what the maximum so we can draw that year's negative 1 here's to Salam only allowed to pick points and I can't tell you what the maximum in that Dinner That's the future here so this is the region them allowing myself to look at here I know there's more parabola chew it but I'm only allowing myself to look at those points on the gravel I I wanna know what the next minute so this yellow rate here is the equivalent of the map in there And how we do this so these are actually pretty easy because We had most called the extreme values there It was the extreme value there and you can see it in effect here you see either the Mac's or the end of the year Turned a critical point that's where it occurs at a critical point or If it does where's the Mac's character will there is no critical point figures as the Max where's the Mac's come from it comes from the boundary points edges so extreme value theory says But the Mac's Death of X on a closed the interval says the setting of the Mac's early in occurs at EU a critical point for a boundary port 2 wires and make it easy I 1st of all you can collect boundary points really easily spread say what those are the Boundary Pointer x equals negative 1 x equals 2
So that's a potential maximum Reddaway I've got those are the only other thing any defined the critical and then don't even need to do any 2nd derivative analysis because I can just take those 3 points that put him in the function ever wants the biggest is the maximum whoever wants the smallest Jets where it's helpful solos
The 2 do that with this problem he said I have had grind of X 2 X said illegal 0 that means that have sleep was 0 and that's a critical point The extreme value theory says my mineral Max is hurry there's a critical point or The boundary for its I've collected All my critical points 1 the boundary points that were x equals negative 1 And x equals 2 So now I have to do to figure out which is the maximum and just plug in those 3 . 5 x equals 0 making candidates x equals 0 X equals negative 1 2 and then I just checked at the VAXes teacher those points of effort 0 0 at a negative 1 and that's 1 of the 2 beds for earlier that lesson I can say immediately with the this is the max and this is the man of the match between Call back The moral the story here is when we wanna do this 1st O'Neill the the critical points And then we wanna see what's happening on the boundary of the other places where you could get a potential Max's
Before we do what we can't see anything because the formulas to complicated will do 1 where we can see everything that's going on
Fajardo extend that up matter for more than attic square post y script reveal use X 1 I want to use those variables as much as possible that problems Booker a assisted direct extension we know what this looks like our a steady few times now that this And is not highway extend this example there well I need to restrict the variable and leaders say that I would say that X wonder between negative 1 and 2 and Let's say said that next to his between native So where that look like ahead 12 get here is a little rectangle goes back into the negative area and I'm only a plug-in points from this rectangle the the way you read this is exited is between negative 1 and 2 next to his between negative 1 some only plugging in these points and which you end up with is you can't picture if they take this again just favors napkin sitting on the table and this is least and not take the corner of the net in and I pull it up here and up with something like this and this little PQV the maximum and on the back side it's also a little thought that got back there As you if you paper Mushayt onto the base you'll see where the maximum and occur but of course that's not satisfied very nice look at it think about but that's not going to give us the answer because we need the algebra to do so you can only do this in this case can see what's going on this is gonna be were the maximum that pulled out up to be the highest point but I need the algebra
To only get that the calculus circuit that them so the extreme value theory and extend amount for the maximum minutes in occur at either a critical pointer boundary point but in this case the boundary is much more more complicated telescope does go scope look at the boundary pillars forget a graph of just the boundaries
Show It's so x 1 with Khaled X 1 and next to those my taxis and They X 1 goes between negative 1 and 2 and X to those between negative ones and to have this picture here that's my rectangle The boundary is that outside of it yellow Ochoa That's the boundary the interior all points on the inside so that's like Over there in between the interval we found a critical point inside their that qualified as 1 of our candidates
But now that the points over the negative 1 of them In the 2 I have this whole founders to check all of that stuff on label the since line 1 line to line 3 online for hooligan after dude room to analyze Maximilian on each 1 of those edges
So that will really go through here the extreme values there says that he there's a critical point on the boundary celestial 1st us go get the critical point That's not hard to do we've done this before so the gradient of death that everyone has to 2 X 1 2 x 2 per cent 0 x 1 0 2 2 0 a weary know that from the picture this is going to be our critical point 0 0 at the bottom of the Travel so we got a critical point Rainier 0 0 villagers say that they year analyze the critical point I just saved it and plugged in theater missile which function value gave me the largest have want critical . 0 They now it look at the edges of this time rectangle square off in the 1st quarter look at is L 1 members is what you have to learn to do is along this segment 1 of the Variables is constant And that's where you start so what what is constant here will What's not constant as X 2 X 2 moving right so as 1 of the things that defines Alwan makes it crystal clear what it is is that X 1 is equal to It's constant and since its constant there what I can do is I can write an alternative version of the function because what's the function doing along this segment for
X 1 is to the whole time so I can plug into into the original Function plug-in X 1 is through the whole time there my new version of ad is gettin be 4 plus X to wear
And Now what happens is I just turned effort into that situation just checking along a segment so I only have 1 variable just do this like we were in matter to a guess what's what's at Prime that's 2 x 2 said if people a 0 Text to neutral 0 so what I get is what I'm really doing is I'm finding critical points On the edges of checking for critical points 1 Casilla but we do we got a critical point on the edge this is X 1 equals to annex to equal 0 that critical in label It said Get X 1 equals J. here it's a little bit Maxtor minimize issue which we could analyze it further looks like it's a noon because You could you can't cavity See positive OK but anyway that's our interest His collecting all the good news is we get to do this for currency you get to to see it and you'll get used to it It that they deal OK L to now or on the top We're and held 2 I want to define what it is X 1 is varying X 1 moved but next to his stuff there 2 of the defining feature of of Tuesday next to it and then we just do the same thing next to his than half become next 1 their last 4 distinct areas that you said it 0 1 I just got a little critical point along that edge and that critical point that 0 2 We could put it in here I've got a little critical point of 3 scale threes over here now and the defining feature is that X 1 is stuck negative 1 race
X 2 is varying X 1 it negatively on the whole time negative 1 now my function f becomes 1 plus next to square
And just go through the same thing as traders bet that said that equal a 0 X 2 0 0 2 . 7 x 1 equals negative 1 6 2 0 critical point along the edge of the glass L 4 Here X 1 is varying that next to his study negative 1 so that F found that 1 squared plus 1 Time is 6 1 2 X 1 0 the same thing over and over in this example it's the same thing over and over in future example of what the weather's good practice with a focus so the 2 points ever collected are 0 4 x 1 and sodomized 5 critical points Along the little square There's 1 extra saying What's the extreme boundary over here in extreme boundary was a negative 1 in the queue over here the extreme boundaries are the quarter-point so I have to add to my list the corner report that steady completely sits I've got 1 2 3 4 5 potential point so so far have 4 more because at that on the extreme corners that At time Group A The corner points are negative 1 native won by 2 2 and negative ones 2 And too You want But I have my 9 candidates there has been a check at his Adidas still half of negative that Siegel To plugging in the X 1 square foot sex square Effort to to that is equal to 8 so far that my maximum depth of wanting to that's equal the 5 and this is also a 5 So far this Guzman maximum mimicking check all these critical points here at 2 0 that's equal before the 0 . 4 per cent of negative 1 0 weekly 1 0 1 1 and F 0 0 0 0 Chris conclusions will see its restricts So x 1 is restricted to that end next to it is also restricted to that box we have a Amax at 2 2 It's a Max was set at 2 2 He will be there so that's the actual maximum value and then we have a minimum 0 0 this was the lowest value that we got here 0 every thing else was positive Yes can't stand at the first-ever yet so when I got L 1 here I ask myself what we which very which variables actually moving along here
It's X to excuse the article variable right next to his moving X 1 is constantly it's equal to 2 the whole time That's why I start off by saying X 1 equals to 1 of arms Anthony constants I find which constant a city people that constant than I played into the function that removes 1 of the variables now becomes a 1 variable problem I find out the maximum of the critical point out of the extra Minn And then I do the same thing here so helps to appear Here x 1 is moving but X 2 is that appeared to start off by saying next to equals to plug not 1 variable problem It's really just a bunch of problems from our to 8 all these urges problems from man to break is just yet You have to do the work depicts the value of the plug it in Many collect all the points is the critical points collect those points at the corner points to the list and see which function value give you It is the biggest and A year in a riot in the streets of class now can get out there hand said Friday it's a long weekend and That's in discussions The rest of the nite air-conditioner saying more 1 of I'll do more It