x smartpost

in multivariable calculus just as insingle variable calculus as soon as we know how to find uh... relative maxima minimal function we certainly are interested even finding the absolute maxima and minimal function because inthat fashion we can solve optimization problems and very similar to how this has gone in single vehiclecalculus

we will talk a little bit about the theory but simple talk for the most part about someexamples and uh... as i think i had promised atthe end of the presentation planes will be talked about distanceswill specifically talked about distances in a separateexample because indian finding the point that his closest to a certain surface for example or that his closeston a certain service to another point

that you something that actually use an optimizationproblem so what do we have we talk likeoptimization about going to talk about uh... largest usefulness values itsmulti variable beat 'cause we're talking about functions of several variables is the optimizationyes uh... we will call unconstrained becausefor constraint multivariable optimization dot use

what we do with lacrosse multipliers andthat will be the next presentation but we start with the definitions thatshowed at least feel familiar name be a number in the is called the absolutemaximum of a multi variable function if elicit d if and only if well first of all thathas to be appoint p and d so that if of pc m in second of all for all other pointsaren t

we have that effort barista sent ripples so basically p ease and input value so that the value with in and that is supposedto be the absolute maximum is consumed by the function the value envious then in fact the largest value that the function assumes

in fact sit if you're interested in the definitionof an absolute minimum well that's a number so that there is any hewlett cites a bit of afew a stick with that number in so that for all other points aren tyou we have the time for congress goodbye greater or equal them dot absolute minimum cell that is not

too hard to formulate int if you could look at this definition if we were just for a minute to forgetthat we're talking about multi variable functions then this piece indeed the definition off the absolute maximum off a singlebearable function that we have already seen in single vehicle calculate so thisdefinition in fact is something

that could work any sitting by the function may take odd may take input values from frombrother stranger sets then subsets of of indimensional space alright so when we talk with absolute maximum wewant to osce certainly have some kind of aninsurance that we can find those things and forthat we have to be a little bit into theory

cell uh... first of all we're going to findwhat abounded said isn't about it's it is a set that is contained in some ballso basically we're looking at the same d here then we can see that if we findourselves a center point we can eight circle around that sandra pointoff a certain radius in of that is the eight ball or a disk in this case basically can use it for which we can dothat in use it that is contained

in some ball in free or higher dimensions or a site eighty sq two dimensions that is what we call the now and that'sit closed sits on the other hand closedsets that good would you describe those daysthey owned the units so basically if we're looking at sets like this one here

where basic pizza solid line settlers of the solids curve here says that this part of thecurve is is also part of the senate in the dotted curve says that this part is not part of theset then set d actually would not be closed andthat is because i could find a courtesy that approaches dotted curve here

we're everything on the kirby's in thesenate but the limit off the curb its not in the second then that would not be considered closed so basically a close to is something that contains every point that thesenate approaches with these two notions week he andformulate the following feo if we have a continuous multi variablefunctions it is to find on the coast and bounded c d then there are points ta beating thecity

so the effort to use the absolutemaximum of f on the domain d in so that there could be is the absolute minimum often wendy eve this theory that sounds familiar to you knowit should be because again if we are just going to you dropped v demand that this is the multivariable function that's pretty much verbatim theformulation

off the corresponding fear m single vehicle captains other challenge of course is that thenotions of closed unbounded are something would require further discussion entthat is typically done in dance classes like really cause of the theory behind this the onthe part what we can also do is we can certainly uh...

weekend certainty assure four cells that at least thehypotheses i needed so for example if we're looking at thefunction f affix implied being expert class whites creek that is definitely acontinuous function on the set dpm uh... are to abuse at two dimensionalplank if thats it certainly is closed becauseof a pretty curve that seats in our two then the limit of the curve will also be a two-dimensional vector so that thingsclose however

the function it has no absolute maximumplenty because of course expert plus was great that grows unbounded actually makeexplain large similarity we're looking at the function of a it'slike being one over expect uswest great that's continuous on a different domain it's certainly notcontinuous at the origin bright because at the origin we divide by zero in thefunction is not defined however is defined on a domain

of all xli sordid zero a smaller thanexpected it's widespread which isn't small rippled and one sothat's what's people call a pointed disk around the origin in the game this thing now is about it however if has no absolute maximum ont_v_ cost yes x it why approached the origin one overexpect less twice that of course gets arbitrarily large even though

the well the ideal ball and it should befairly simple to understand the idea of credo was basically means you can't have thesekinds of calls that we see you here and that's what we're going to be thatat for the purposes off the first multi variable calculuscourse these two hypotheses really must besatisfied otherwise this theorem will not we're not going to produce the urineco-starred is something that certain he

requires beeper machinery but that means we are going to start with examples assoon as we remind ourselves of what the process of finding out thatmaximize well what we need to do it we need to find a critical pointsbecause those could be better than blacks or minimal we need to find the values of thefunction at those critical points that will be no which your relativebig-screen marv really high torque really know we need to find the largestvalue of the boundary

in the largest of these numbers use ofthe absolute maximum slow the process of finding these things exactly the same as for single variablecalculus only fax the computations will be more complicated because for critical pointswill need to work on the gradient and the boundary ofcourse now will be aboundary curve or maybe even a boundary surface so that will take some somelonger one thing that we

i want to talk about here is what's the bombing we haven't fully defined uh... it's it'snot that hard to do really basically it's going back to that picture that ishowed you when i talked about closed sets let'stake a look at this it int uh... it would be very natural to coursesay that this blue car if you use the boundary inspire that within means at this pointp here visually

is all of the boundary and what does that mean that means i canapproach that point from the inside office at the curbc_-one in from the outside office at the curbsitu if i want to formulate thatmathematically keenly i would say that if i have to sit d_e_a_in individual states actually again we're not going to limit ourselves to two dimensions threementions three dimensions or anything like that

uh... these notions are all easily generalized up in the pointy is said to be a sex point on the boundary of d if and only if there are vector valuedfunctions gamma want to defined on zero once again no one wouldbe the parametrization of c_-one gala to would be the parametrizationc_-two and those would have to be so that for all t

get all of us to use the indy that'sit's what for altigen majority is not indeed into limit s were approaching the point that is not in the domain offthe curve forget all one of two uses the put btgoing to zero in this fashion that is pete int if gamma too approaches the left in office

uh... domain if he approaches zero four gamma too siebel seed to what also approached thepoint p so that means that weekend approached any boundary point curves that are entirely outside in entirely inside the domain in top with other boundary point then

is itself in the sixty or not that depends on what the deal is closed okay now we can take a look at someexample so let us find the absolute maximum off thefunction if of xli being three x squared plus twox y minus y_ squared on it right in that was vertices zerozero one one is negative what one let's visualize that first

we have x-axis we have a lot axis viewsthe origin zero zero here's two point one want here's the point negative one one intent we are interested in them absolute maximum with this function one piece triangle here well that means we first have to figureout if there are any extreme of the east side of the triangle

and then we also have to find out how the function behaves or in fact boundary well and dot shouldn't be too bad because all theboundary we can just plucking uh... linear parametrization characterizations nineseconds uh... first of all we want to find therelative extrema overall so we take the gradient office

insect that equal to zero well the partof the ridable f with respect to x is six explosion bite and that needs to be equal to zero is the part of the radical stance withrespect to apply too xd minus two y which is supposed to be equal to zerowell-thought tells us very quickly that why is he quips x in a free plans thatinto the fact that zero is supposed to be six explodes too well i we end up with him uh... because fixesequipped why we end up with six x two x

which is any excellent access to thezero that tells us that x is supposed to beput in zero percent tells us that by is supposed to be equal to zero is that these the only critical point off the function now itjust doesn't seem to have a calculus we have to make sure uh... with at this point actually useinto domain and of course it is it is in fact on theboundary so technically we should check thatlater when we talk about the boundary

but either way we might as well work it out here if ofzero zero is equal to zero if we would have madeat that point for the boundary neatly now if this point had not been insidethe domain just as in single variable calculus we would have needed to discard point because it is not relevant for thedomain under investigation it would've been

not relevant for the point underinvestigation for the boundary curves well basically we've got the horizontal line group i a sequel toblend well then why is equal to one int then exist between negative twenty andone uh... we would need to investigate the croat f of extnone which will be three x creek plus two xminus one and that's occur for which we can justfine ther

uh... extremal with regular singlevariable calculus i take the derivatives with respect to x insect that equal to zero and we end up with what six extras towhich gives us pics equals negative wants her and so that means odd because negative one-third isbetween negative one in one that's the point that we should investigate instill because well i was supposed tobe for the one we need to find out

what's if of negative one-thirty onenessand that ends up being three times one ninth which is one third class to would times negative ones foraccounts one which is minus two thirds of get minus one third minus one instead minus four thirty anything here of course we also have tofigure out what info one one is n_f_o_ one one use three-plus two-minuswants four if we have to find out their company ifone wanted to st minus two minus one

which he's zero so so far the absolute maximum yes for ins you can go with the question doesn'task for at the oxford minimum actually needed for thurs uh... let's take a look at the otherpieces uh... y equals x intend exp green zerotwo point that's the up still off the train on the right side well that would mean

that we have to find out vote thefunction if affix the index that's right because uh... wise equal to explore thatfunction is forty ext squared adds that assumes its minimum at zero and the origin has been already processto get enough to worry about that and the maximum is assumed x equals one and one one has beenalready processed so

we've really have known you information from this part off the triangle in if we're looking atwhiteness minus six intent native once more a quick smart post herethat would be the downward sloping side of the triangle in the picture thatwe had well then we have to figure out

what careful exp minus sixty s n_f_o_ exit minus its fees uh... cedar would be three x squared minus two expert minus xsquared so that would be zero and that means we also to give any new information on dockside and so that means the absolute maximumvalue is the value air force one one which is people to four

if we want to visualize that heeey iss the function workout uh... in mask and we can see that first of all pick is certainty defined on a triangular domain right inthis case now the white axes acted going to the side because remember that

in these three-d_ pictures we always hadthe x-axis like this one in the white axis going that way if we can see that the maximum resistantone one and we can also see if we want to that'spokey wickedly hot on one of the stopping sides of the triangle functionto play forex squared past that part of theother side it was horizontal zero that's this part is or the third side we actually havefound uh... that there would be an absoluteminimum assume which we can also

cds pictures self it looks like we have done this problem correctly bns we wanted to record for completenessthat you can do is not asked for in the probably could also say that the absolute minimum is that i did make it at four thirty so basically set the grading equal to zero and then

deal with the boundary which means we could used this now toactually also self an optimized asian problem that really are a bit familiar with still from us into their book after this intensivewe want to look at impact and get a box without a top that is supposed to have volume fiftycubic inches and we want to find the dimensions oftalks

with the smallest surface area this is a problem that uh... you'll probably have seen instandardized uh... in single variable calculushowever in single variable calculus we always have to demand dot the blocks had a square base which actually is reasonable if you want to find a boxwith the smallest surface area with multivariable calculus were actuallygoing to show not only that it's

reasonable but that it really is the only way the smallest surface area so this timearound we've really have a box what we have three dimensions and debts a body intensity and so we're not going to assume thatit's a cyclical by a sweet used to do in

single variable calculus well it now in order to uh... finds most surface area when the volume is given what we need to the volume equation service your anyquestions of the volume six times by time seemed short which times dates times heights is are that's supposed to be fifty cubicinches in the surface period is eight whilstx_y_ that's the area of the bottom

costume time sixty votes right panel here index left panel yet just two times when i see it that is front panel yet the area off the back painful okay so that gives us in area as a fontfunction of three variables let multivariable calculus

we shouldn't be bothered by that anymore but since wehave an equation that relates to three variables to eachother we can eliminate one of them anti-western and eliminates the becauseof course from the volume equation we see that z diskeeper two fifty cubicinches divided by x y so that means area of exited why surveyor of x impliesex-wife plus two x times z which is fifty overex-wife as to why

times fifty over x_y_ and so that wouldbe x_y_ plus one hundred over why plus one hundred over x and that speak sake what we have here the domain of course fees that x inquireare spokes supposed to be greater than zero anti the rest is just computation take our surface immune function and what do we need to do we need to

find a gradient said to be put to sea sowe that possibility with respect to expquickly zero and that would give us why minus onehundred over x squared when we take the derivative with respectto x we take the partial birth rest to why isaid that equal to zero twelve that gives us a partial derivative obviate x here atthe credible x_ y_ with respect to wise x and the drivel one hand of one hundredover why with respect to boys made one

hundred over one square interpreter for one hundred or x withrespect to why that's just straight zero syndicate x minus one hundred who one hundred overwhat square that means that exclude wise i'm drivenby its critics is a hundred expert was a hundred works critics is a hundred if these three questions we justtransfer over and now we have to solve they system off too non-linear equations in

two variables and as i always say for noted in your equations there he'snosair alireza talk to solve those however when we only have twentyquestions for two variables tiki the best thing to do with salt one equationfor one of the bay area bles implanted into the other equation so here i think outsole for why probably sure at if i solve this first equation for why i did why he calls one hundred over

if i saw this solve for x okay if i saw this certainlyquestion for excited exp quills funnel one hundred over why squared in effect like they've been here forexcited yes squared from right appearing ten times why hereequals a hundred and that's this one hundred magic all right so now i have

one equation for white landscaping populous states here againto you uh... also simulate the situation that you havewhen you're being advanced text people will skip steps what do i have you have got a hundredsquare on the left side of got a hundred on the right side kids lunberg end upwith a hundred on the left side and i have to live over white to thefourth on the left which is one over whites you can i can multiplythrough by cuba over signed up with

white cubes equals one hundred that's easy salt that tells me that whyis the third route off one hundred if i could access back in here for whysquared i realized that x times one hundred totwo-thirds is one hundred and if i cancel a hundred to two-thirdsi end up with x equals one hundred to the one-third

and that means that x is that there are four of one hundredalso and obscene then of course is fifty over ex-cons lie so that would be fifty overthe third group of one hundred square it in fact would mean uh... because fifty is of course halfoff a hundred i can multiply top and bottom with too that gives me a hundred on

on talking to one half hour front and ahundred over slayer of one hunt uh... a hundred over one hundred to the two-thirds of thehundred to the one-third in fact he stood for agroup of four hundred services one-half for route sont v only critical point

we can't find ispat ache since people tobody and so in terrible secret service the corresponding vertical dimension is one half times thethird route off one hundred now we also have tocheck the boundary here and basically the boundaries that boundary of thefirst quarter and so it's not function is ko xli equals ex-wife plus onehundred over why plus one hundred over x that functiongoes to infinity when ex-wife goes to infinity

or when x and y out what one of them goes to zero cidanytime we approach the boundary off the first quarter whichwould mean if you want to look at an infiniteboundary that something goes off to infinity in which case of or six times like wasoff to infinity or something approach is one of the axesin which case explore why goes to zero in that case this one or this one willgo to infinity so this is a function

builds up on all sides and that means this critical point that we found must be absolute minimum length that means thedimensions off this box with the smallest surfacearea really are for a group of one hundred in interesttouch the roof one hundred interest times one-half sort of one hundred impinges i didn't work out

the decimals for that are because if you've got to play out likethis you see very easy to use it the volume previous fifty cubic inchesleaves you also cvd easily that thing that this thing must have a square vaseinsane this last part here shows that the height is actually half off that we thought that

okay we finish we've the computational dispensers in the example that i have here he is that i want us to find a shortestdistance between two point zero one two in the program await z_ equals exclude plus points with ifyou visualize that i think that what they want you to dothat yourself i'm not going to show the picture right now obligated visualize that their actuallyis a way to

trick your way through this problem andsolve it with single variable calculus however i want us to look and and i'm going to tell you how to do the at the end wouldbe to another visualization but i want us to just realized thatdismiss computations are something opt for which in this kind of optimizationis really pretty much made for

because all we need to know in order tofind these kinds of distances fees the distance for me that in the processof finding absent maximal or in this case absoluteminimum the distance between two point x_y_ and z_ o_ x_y_ orin fact probably in two point zero one two oh well that'sgiven by the distance for me that is to square root of the different off the square of thedifference being that

x components so x one zero quantitysquare class the square of the distance in theuae components white minus one point square uh... classes at the square off thedistance in this e components z minus too quantity squared and of course physioxli is exclude class was great so we can cutthat in it's minus zero quantity statistics squared by last month'smeetings with us intently half exquisite classify squared

right here uh... minus two quantity in so basically to find the shortest distancebetween this point in desperado look we have to minimize dysfunction that is why distance computations really ultimatelyi just dont

multi variable calculus because we canjust set up the distance for me to intent find the minimum with calculus the annoying part here is of course thatwe've got the square root and it turns out that if we were tominimize that squared function ki wo theck slidingexcluded wide minus one quantity square plasticscriptlance west replies to quantities creek gives the square off the minimumdistance in fact is

just as well because then if i want tofind the shortest distance i would just go ahead and take thesquare root of the so we're actually going to focus on dysfunction kyuki taking dysfunction q we need to find where degrading to zero so i need tofind of the part of read with q with respect x well that's true x from here the partial off one minus one squarewith respect to exit zero insane

the partial three decks grade class wasgreat minus two quantity squared with respect to exist to xref class westward minus two times to extent that's supposed to be zero theculture of q with respect to why well the derivative of expert withrespect to wise zero the derivative of why minus one quantitysquared uh... tool by minus one when wedifferentiate with respect to why

into derivative expert class wisecrack minus twoquantity squared uh... is tour times what every since iexpect us why screwed minus two times did read the book inside with respect to a wide which is why and that's supposed to be equal tozero also and so we are again danced with glee system off to numbing equations

for two the rebels this one is a little bit rougher than the other one so let'sfirst just take a look at the first equation if i cancel it to a year i end up withexpressed two times expert plus my script minus two times x equal zero and i can see here for a factor out in xend up with x as one plus two more times expert passmy script minus two

equal zero in fact gives me on one hintthat x would have to be equal to zero or that's this parenthesis has to be put in zeroin the strength of this is basically to exclude class to why squared minus four plus one so minus three so thatwould mean that x to expect oswestry would be good for three and i think thatexpect

class was great equals three hats we're going to do now is still going to solve this system of the questions first in this situation been exiting quicklyzero insane in the situation with exclude plus what's great is equal to so let's go ahead and saw this thing ithink we are first going to consider the

case that ecstasy zero well we have to see other equation whichis to why minus one plus to expect less weisberg minus two times to wine equals zero and that of course with its equal thirtycomes to one minus one plus two white script minus two times to oneequals zero which is dropped the exquisite right because we don't have to carry a zero with usthere but i can slip through again

look at my minus one class wisecrack minus two times to what equalszero all right and so now if a multiplierparentheses were to begin tweak it to white house was quickly which is towhite cubes we get negative worldwide class lights minus three y and minus one so we had to wake you minus three yminus one now equations like these are in in some ways attract beacause i haveseen students usa political you know in something likethat and i'm i'm not happy about it and

i'm not trying to make fun of people butof course that doesn't work because this thing he's a cubic equation a quadraticequation it's just one of those things where there is the last temptation thatwe should not tragedy yourselves into it so how do you sell the cubic equation well basically uh... their easter cubic formula whichwhich we don't memorized

because it is just a little bit too complicated you can weirded out uh... numerically so you canuse the solver or sometimes you can't use the rationalzeros theory if you're lucky enough that there are uh... certain rational zeros anddysfunction actually has his uruguay equals negative one because two attempts made one cuba'snegative to minus three times negative one would be classed reinvent minus onceyou get minus two

plus three months one that is zero so that means that this polynomial must have a factorby class one in if you've been do your synthetic division u realized that the remaining quadratic asto why squared minus two y minus one into again if u ravine something like this

in a takes well the least you can do is verify that this really is correct yourealize how people found the factorization is the fact that it iscorrect yes easily verified by saying that a key tomy spirit times why is the two white cute and wehave one times to white square at minus two white times lie that's too was great minus two ways forits within the white square terror

and we have one times negative two y minus one times why that's the linestream going to be one-time snagged one which is the one so that means on one hand we have thatone one equals one is a solution in for the other one nowwe can use the quebec for weekend so we get negative be negative negative two-plustwo-minus be squared four minus forty see someone is four timestwo times that one which is

classy divide platform and uh... then if u factor that outsquare root of four cuz its grip twelve is to terms with three if you cancel the tombs you end up withone plus minus four three in those numbers are approximately onepoint three six six zero native point three six six zero some of those farsi x value zero at the white value oneextended zero point value approximately one point three six six

index value zero byte valueapproximately native form three three three six six uh... those are three candidates for the point where the shortestdistance between the point and the probably is realized is although we still had the case

that extra class was great equals threehaps again we look at this other equation which wasto by minus one plus two times expect was my script minus two times to white equals zero in if we could xc recross quest forequal street have skiing here we end up with a peek into the tworightly why minus one this too was also canceled and we have it's great becausewhat's great being three has a minus these two times to why will three guests tries to isone-half one half times to was once we

end up with wine might minus one minusline equals zero that gives us negative one equals zerowhich has no solution so that means there are no further solutions to thissystem of to be questions in to variables and well so now remember that the distance formula isspirit it's clear class one minus one squareclass it's correct oswestry plans toquantities quick to dismiss florida for

this problem we cannot let you zero eight negatives one in fact ends upbeing a distance of route five that's something we can still compute directlycause begets zero we get monies to spread which is four and we get one minus two qualities creekwhich is once a sort of focus on just sort of five certifies approximately two point twothree six one

we also have to compute the distance uh... that's bs realized that exp qualzero and white equals one minus route three overto you and that's something that talk straight into a calculator it is present that being approximatelytwo point three one two six in the distance realized for x equals zero employee costone classroom tree over too dot ends up being

zero point three eight nine seven so that's just computer calculation i wouldn't want usto do any kinds of acrobatics with square roots and squares for those things and so that means the last one is theshortest distance the obvious because that's the smallest number and now let's look at visualization ofthe whole thing because it's kinda interesting

dot apparently we could have today uh... shortest distance off to a point two three six one in somesense uh... eighty further distance two pointthree one two six invent the shortest distance being pointthree eight nine seven and basically the visualization goes back to what imentioned we're here on

uh... that this problem actually click uh... tricked into becoming a wonderfulproblem in fact it's because when we are looking at the point is zeroone two int a rotationally symmetrical object at the problem of course isexactly this parabola rotated around is the access my it is normal to conclude that uh... exported off the point where theshortest distance is realized

must be zero because of a point doesn't have a cscoordinate zeros and i could make that exported zero closer to with the probably that's because the geometry is rotationally symmetricbut that with the means i would just have to look at that shortest distance between two pointby equals one z equals two

parabola zine equals whites creek whichis just any slice parabola weight along with the y axis or at the intersection of the problems withthe wise the plane and so now let's take a look at this animation onemore time as it comes around low-key so this

is the cross-section off the parabolaweight we've the wise the plane why does horizontally z goes vertically here's the point why equals one z_ equals two andremember that now x's equal to zero right and if you want to see the wholethree-dimensional thing in your mind you would have to

imagines that this is rotate around is the access to give us akerala the distance to find out just eighty point on this kerala or parabolalloyd and later to run along the whole gamut off this parabola i can compute the distance every single one of those points okay sothat's that a crime and at the first point where z animationstops

i have a distance of two point two orthree six we chase this business here and that is locally issuance dismissed because itturns out that it's like a little bit to the left acute two point two three seven uh... intent to point to seven and so on so thatthat's that

doesn't go to the left i get a larger dismissed if by stepping a little to the right come on this all yet says the policy at if a step up to the right and also get paid larger dishes so that you don't trees insuch a break at this point here is closest

in this period but not overall because the distance now is increasing and the distance is increasing up untilthe two point three one two six oh two point three one three here because thisthing is rounded two three digits so this is now alocally farthest away points in if i step into the left little to theright from this point

that i'm actually going to get closer tothe group point again which at least overstepping to the rightas it's happening right now is quite obvious this there's a whole lot closer than that and so this distance now that point three ninety four point threeeight nine seven is the closed as these things andbecause as we go past this point now we can alllegal farther away

that shortest distance between two point one two and theprobable equities that approximately point three eight nine seven that's we have does it for this presentation basically what would you always want todo is you certainly want to be able to see what's going on i think you want tobe able to make sense all of the results just as wealso have done here with this

cross-section if you find out in the process thatmaybe there's a way to solve this problem with single variable calculus you can use it as a double check buteven the hint if you've solved the problem with a ballot method and youhave the solution and that is the main thing and what we have been seen also hearingthese examples is the power of these computations becauseaslo as we work through these computations correctly

everything it's going to work out justfine question drawback is we have to no engineer systems off equations whichhe has something for which as you can see here out there is a variety of approaches andoften it's just a matter of pattern recognition finding a weak spot finding somethingthat allows you to simplifies the equation sport worst-case scenarios that you just sold for one vehicle and then plug thatinto the

other ones that is something that ididn't do for this last example because solving cubicequations or solving quadratic equations typically leads to adhere expressions than i care to work with for thisexample but that's where that factorization that we saw actuallyhelped us very nicely alright so that means for the homeworkwe all you have to do is set the grading equal to zero check the boundary move onor you're taking the boundary for this example of course

the boundary it would be that x and ygoes to infinity and then the distance goes to infinity so that was not anissue there except the main equal to zero check theboundary find the largest for the smallest value depending on whetheryou're looking for a maximum or minimum hopefully not getting to bothered by thecomputations in between alright have fun with the home