## Math for Economists - Lecture 12                     Video in TIB AV-Portal: Math for Economists - Lecture 12

 Title Math for Economists - Lecture 12 Title of Series Math for Economists Part Number 12 Number of Parts 15 Author License CC Attribution - ShareAlike 3.0 Unported:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. Identifiers 10.5446/12906 (DOI) Publisher Release Date 2013 Language English

 Subject Area Mathematics 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
Point (geometry) Statistical hypothesis testing Definite quadratic form Greatest element Functional (mathematics) Multiplication sign Maxima and minima Wave Derivation (linguistics) Kritischer Punkt <Mathematik> Lecture/Conference Term (mathematics) Negative number Matrix (mathematics) Hill differential equation Position operator Convex function Faster-than-light Gradient Maxima and minima Multilateration Calculus Variable (mathematics) Thermodynamic equilibrium Inflection point General relativity Right angle
Definite quadratic form Point (geometry) Connectivity (graph theory) Equaliser (mathematics) Real number Multiplication sign Mereology Derivation (linguistics) Kritischer Punkt <Mathematik> Lecture/Conference Forest Matrix (mathematics) Negative number Nichtlineares Gleichungssystem Determinant Physical system Social class Eigenvalues and eigenvectors Gradient Mathematical analysis Linear algebra Calculus Variable (mathematics) Vector space Combinatory logic Directed graph
Point (geometry) Functional (mathematics) Kritischer Punkt <Mathematik> Matching (graph theory) Lecture/Conference Maxima and minima Flow separation Sinc function
Complex (psychology) Linear equation System of linear equations Functional (mathematics) Gradient Calculus Variable (mathematics) Substitute good Derivation (linguistics) Lecture/Conference Term (mathematics) Square number Matrix (mathematics) Right angle Nichtlineares Gleichungssystem Resultant Physical system Social class
Lecture/Conference Multiplication sign Oval Student's t-test
Point (geometry) Focus (optics) Functional (mathematics) Forcing (mathematics) Multiplication sign Maxima and minima Line (geometry) Fisher information Derivation (linguistics) Kritischer Punkt <Mathematik> Lecture/Conference Different (Kate Ryan album) Negative number Nichtlineares Gleichungssystem Physical system
Point (geometry) Inflection point Kritischer Punkt <Mathematik> Lecture/Conference Matrix (mathematics) Maxima and minima Series (mathematics)
Point (geometry) Group action Graph (mathematics) Scaling (geometry) Multiplication sign Direction (geometry) Range (statistics) Interior (topology) Maxima and minima Equivalence relation Inflection point Mathematics Lecture/Conference Profil (magazine) Analogy Saddle point Chromosomal crossover Right angle Figurate number Directed graph
Point (geometry) Functional (mathematics) Graph (mathematics) Eigenvalues and eigenvectors Direction (geometry) Calculus Mereology Variable (mathematics) Flow separation Derivation (linguistics) Prime ideal Kritischer Punkt <Mathematik> Causality Lecture/Conference Different (Kate Ryan album) Oval Universe (mathematics) Saddle point Negative number Modulform Determinant Bounded variation Position operator Resultant
Metre Dependent and independent variables Multiplication sign Direction (geometry) Mereology Derivation (linguistics) Kritischer Punkt <Mathematik> Positional notation Lecture/Conference Term (mathematics) Order (biology) Saddle point Determinant Position operator Spacetime
Point (geometry) Radical (chemistry) Focus (optics) Lecture/Conference Term (mathematics) Direction (geometry) Interior (topology) Negative number Saddle point Maxima and minima Determinant Product (business)
Point (geometry) Functional (mathematics) 1 (number) Derivation (linguistics) Mathematics Kritischer Punkt <Mathematik> Lecture/Conference Term (mathematics) Saddle point Matrix (mathematics) Nichtlineares Gleichungssystem Diagonal matrix Determinant Extension (kinesiology) Algebra Physical system Convex function Process (computing) Eigenvalues and eigenvectors Point (geometry) Maxima and minima Line (geometry) Variable (mathematics) Degree (graph theory) Proof theory Radical (chemistry) Right angle Directed graph
Point (geometry) Functional (mathematics) Group action Maxima and minima Mortality rate Food energy Equivalence relation Differential geometry Parabola Mathematics Kritischer Punkt <Mathematik> Lecture/Conference Order (biology) Extremwertstatistik Energy level Boundary value problem Figurate number Set theory
Point (geometry) Functional (mathematics) Kritischer Punkt <Mathematik> Matching (graph theory) Lecture/Conference Mathematical analysis Extremwertstatistik Boundary value problem Maxima and minima Musical ensemble Vector potential
Point (geometry) Area Multiplication sign Maxima and minima Rectangle Variable (mathematics) Lecture/Conference Well-formed formula Square number Negative number Table (information) Extension (kinesiology) Algebra
Point (geometry) Graph (mathematics) Kritischer Punkt <Mathematik> Lecture/Conference Interior (topology) Extremwertstatistik 1 (number) Boundary value problem Maxima and minima Calculus Extension (kinesiology) Rectangle
Point (geometry) Functional (mathematics) Greatest element Multiplication sign Gradient Maxima and minima Line (geometry) Rectangle Variable (mathematics) Celestial sphere Kritischer Punkt <Mathematik> Lecture/Conference Square number Boundary value problem
Area Prime ideal Functional (mathematics) Scaling (geometry) Kritischer Punkt <Mathematik> Lecture/Conference Multiplication sign 3 (number)
Point (geometry) Functional (mathematics) Focus (optics) Group action Observational study Multiplication sign Equaliser (mathematics) 1 (number) Maxima and minima Variable (mathematics) Explosion Goodness of fit Kritischer Punkt <Mathematik> Lecture/Conference Square number Queue (abstract data type) Cuboid Boundary value problem Position operator
Point (geometry) Functional (mathematics) Kritischer Punkt <Mathematik> Lecture/Conference Multiplication sign Maxima and minima Right angle Variable (mathematics) Social class
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
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
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
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 hidden