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Find the critical points of the following function. Use theSecond Derivative Test to determine (if possible) whether eachcritical point corresponds to a local maxim...

Question

Find the critical points of the following function. Use theSecond Derivative Test to determine (if possible) whether eachcritical point corresponds to a local maximum, local minimum, orsaddle point. Confirm your results using a graphing utility.f(x,y)= 6y e^x-6e^yThe critical point arethe local maximum areThe local minimum areThe saddle points are

Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. f(x,y)= 6y e^x-6e^y The critical point are the local maximum are The local minimum are The saddle points are



Answers

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $$f(x, y)=x y e^{-x-y}$$

Yeah, he said For number 39 the function given is f X com away called why it is the power X minus. It is the power wife. First of all, we need to find the critical points for this. We need to differentiate the function with respect to X partially first and equality equal to zero. So why will be taken is constant. So this will be why it is to the power X equal to zero. But it is the power X cannot be equal to zero. So why is equal to zero again? We have to differentiate the function with respect to y partially and they quit it equal to zero. So ah, here. Why will actually be taken at Constant? So this will be eat. Is power X minus? It is to the power y equal to zero, which means it is to the power X. It is the paddocks, um, equal toe. It is to the power way. Okay, Now let us plug in this value off. Why here? So we will be getting it. Is the power X equal to it? Is the power zero equal to one so we can say x equal toe zero. So we have political point. Only one critical point. That is, uh, zero comma zero to see the critical point. Now differentiate F X, which is effects. Why it is to the products. Yeah, why? It is the products partially with respect to X. Again, this will be xx equal toe. Why it is the products. So at a critical point zero comma zero, this will be called to zero. Similarly, we have a fight culture f y. I call it the way to the power X minus he to the power. Why? Due to the power X mindset bar, Why so f y why he called? So this will be equal to zero and simply we'll be having minuses about away. So at a critical point zero command zero, this will be equal to minus it is the power zero which is minus one. Okay, now we have to differentiate this value with respect to y. So we have to differentiate this. That's it. That is partial derivative off first, partial derivative off the function with respect to y so, which means we need to find the value off. We went to find the value off f X Y, which is curly by Curly. Why effects, which is called liver Carly. Why? Why into the products. So this quantity will be taken as constant. It is the Power X now at zero comma zero. That is a critical point. This will be equal created for zero which is equal to one so discriminate. And at our zero comma, zero will be f x x at zero comma, zero f y y at zero comma zero minus except extra at zero comma. Zero hold square. So we have will be having zero in two zero in two. Because this is zero. This is minus 10 into minus one. Okay, so we will be having this as zero into minus one minus one whole square. So this is minus one. Okay. Why did the products? Why? Into e to the Power X. Now we have be as negative discrimination and as negative, which is minus one. So F has a saddle point. So the function has a several point at zero comma zero. That is at the critical point. We have a saddle point

This is problem number 36. Given function is F X, come away equal to X minus y by access square plus y squared. Okay, so we need to first get to the critical point. Okay, So for this, we have to differentiate partially dysfunction with respecto X. So keeping y is constant so and equated equal to zero. So this will be access choir. Plus why, Squire in two. Our differentiation of X minus one with respect works is one minus X minus one. Uh, this will be to works. And for wise quite with zero. So now we tried this. My access choir plus y squared hold squared equal to zero. It means this is X squared plus y squared minus two x squared. Plus two x bye access choir. Plus why Squire? Equal to zero. Okay, which is why squared minus X squared. Plus two x by access squired plus y squared. It's quite equal to zero. Now. If he just cross multiply, we'll be getting why squared minus X squared plus two x equal to zero. Okay. No differentiating the same, uh, function with respect to why partially we will be getting and equating equals zero Of course, you will be getting access. Choirs bless wire square access Quite persuasive on letters. First, write it. Question number one X purpose. Why square in 20? Because differentiation off X minus one with respecto Why only will become zero? Because there is no trans containing why in the numerator minus X minus one into two I divide by access choir plus y square, whole square, equal to zero. Okay, which means this is minus two X y plus two y equal to zero. Okay, let us write access square place. Why square holds court also now, after cross multiplying, this will be okay. This will be let's take minus two were common. It was already common X minus one equal to zero. So after solving this, we will be having X equal to one. And why I call to zero now when x equal to one. So a question number one from a question number one, which is why square minus X squared plus two x Why Squire minus X squared plus two x equal to zero. So we will live Plug in X equal to one here. This will be why square minus when plus two equal to zero. So why squared? Plus one equals Use it or not possible? Not possible. Okay, Now, if you plug in Mexico toe Sorry by a call to zero in this equation, we will be having minus X squared plus two x equal to zero. So if you take it as common, we'll be having toe minus X equal to zero. So we have to Values of X one is X equal to zero and x equal toe to So there are three that the two critical points two critical points are zero comma zero and ah, two comma Zito. So these are the critical points. Now let us, um we have we should write first effects with specter different parts of the transition off the function of respecto X We have y Squire minus X squared plus two x Why square minus X squared plus two works by access squired plus y squared whole square. Now we need to differentiate this partially again. With respect to X, we will be getting access choir plus y Squire, hold square in two differentiation off on the numerator with respect to x zero because I will be taken as constant minus two works plus two minus. Why? Squire mine, Sexy Squire plus two works in tow to access squared plus Y squared into two weeks. Divide by access squared plus Y squared. It is to the powerful Now value off this F X this develop transition with two uh, x at first critical point, that is 00 So this will be 00 000 So yeah. Mhm. Okay, this is X squared Y squared. Okay, his So this isn't defined again at two comma zero value of this x x at two comma zero will be Go to the fourth for four square. That is 16. 16 in two minus four. Tourism miles four to minus four. That is minus four after that minus. Okay. Why? I called zero X equal to execute toe too. So this will become zero. Everything will become zero. Yeah. Divided by four to the power four. This will be minus one by four. Okay. No, we already have f. Why? That is the partial differentiation of functions given function. With respect to y we have this, I'm ok. Minus two x y plus two y minus two x y minus two x y plus two y divided by access choir plus y squared whole school. So we have not to find. We have to appreciate this partially with Inspector. Why again? So this will become Access Square plus Y squared, old Squire. Okay, this picture wire, this will be minus two X. Let's to minus minus two x y plus two y in tow to access squared plus y squared into do what developed by mhm X squared plus y squared, raised to the powerful. Okay, now, value of this at zero comma. Zero value of this at first critical point, that is zero comma zero will be 000 Everything will be 00 by zero, which is undefined. Okay, now value off this at two. Comma zero. That is second critical point at second critical point. This will be so. It was a four Squire for squired and to minus. For that is minus two. This will be 00 so everything will be zero. Divide by to raise to the power to raise to the power. It will be, uh, do that for four days above four. Okay. Four days of the path for that is minus one by eight. Don't. Okay, so 16 16 minus four. Two miners, four watts minus two. It should have been minus two. Okay, minus tool. So they should have been won by eight. Minus one by eight. Okay, No, let us one second. Right equal toe. Why? Square minus X squared. Plus two works. Why? Squire Mine. Sexy Squire Plus two works plus X squared plus y squared. Hold square. So differentiate this partially with respect. Why? This will be why square study access choir plus y square, whole square. Differentiating off the differences of this with respect, why will be do I only minus y squared minus X squared plus two works into two x squared plus y squared into two I divide by. That's a Squire plus y squared raised the bar for so value of this at a critical point to common zero two comma zero will be 24 the 16 0 Zito. Okay, so zero by, uh, four days is about four. That is zero. So we have discrimination. And at two comma zero. Uh huh. F x x at two comma zero into f y y at two comma zero minus f X way at two comma, zero whole square. So this is minus one by eight and two minus one by eight, minus zero. So one by 64 this is one by 64. So now we have certain things discriminate. And at two comma zero equal to one by 64 and f x x at two. Comma zero is equal toe eyes equal to minus one by eight. Which means this is positive. This is negative. So at two comma zero, we have local Maxima at two comma zero. Okay. Thank you.

We have a number 35. The function given is X over one place access choir plus y squared. So first we need to get the critical points for that. We need to differentiate this function partially with respect to X and equated equal to zero. So this is in the form off you buy we so it's partial differentiation will be one plus X squared plus y squared into differentiation Off one is one Sorry, access one minus. We will be keeping X as it is and differentiation off this quantity with respect toe ex keeping wise constant. So two weeks he went by one plus x squared plus y square in tow, Whole square, equal to zero. So, uh, if you cross multiply and opening the back it this will be one plus x squared plus y squared minus two access quite equal to zero, which means one minus X squared plus y squared equal to zero. This is a very question number one. Similarly, we have to differentiate the function partially with respect to y, and it created equal toe zero. So, no, this will be one plus x square plus y square in +20 because we have X in the numerator and with inspector y axis constant. So this will be zero differentiation of X with respect to y zero minus x will demand as it is one plus. I'm sorry how we have to differentiate this So one will have zero differentiation. Placido plus two y divided by one place Access square plus y square, Whole square, equal to zero. Okay, so now we will be having two x y called to zero so X equal to zero. And why I called to zero. Okay. And we have a question. Number 11 minus access squirrel plus y squared equal to zero. So let us plug in. Let us Ah, look at this. When x equal to zero, this will become one minus zero plus y squared equal to zero. So one plus y squared equal to zero. Not possible. Okay. And when why call to zero? We should plug in here. So one of minors access choir plus zero equal to zero. So x will be called toe plus minus one. So we'll be having two critical points when X is minus one y zero when X is one wise zero. So these are the critical points. Okay, No. We will do the world differences in, or we can say partially transition off FX with respect to X. Our double partial differentiation off FX four with respect to X. So we have to differentiate this quantity. Okay. And this content is one minus x squared plus y squared by mhm. This is one minus X squared plus y squared by one plus x squared. That's why a square hole square. We have to differentiate with respect to X. So this will be one plus x square plus wire square. This will be taken as it is and differentiation off the numerator with respect works. So minus two x minus, we will write it as it is and differentiation off the denominator. So this will be two in tow. One plus access choir plus y squared in 22 x, he right by one plus access choir plus y squared whole square. Okay. No, we'll be plugging in the values. Okay. Now we'll be finding value off this expression at minus one. Comma zito so x equal to minus one. This will become one plus 12 to square into tau minus one minus 100 So this whole thing will become zero. Okay, so we have only this divide by the C one plus 12 square that is to square. So this will get canceled out. So when you is to similarly uh huh. This is too similarly value off the same quantity. Same expression with respect at one common zero. So well, we will be one plus 1 to 2 square in two minus two, and this will become zero. Divide by two square. So this is minus two couple. Similarly, we have to find double differentiation. Possible partial differentiation. With respect to why so simply we have to differentiate this expression with respect away. So after, uh, solving this will be getting minus two x y by one place successful plus wise under minus two x y. We have to do Carly bike Oliva minus two x y by one place access choir plus y squared whole square. Okay, so minus two will get canceled minus two will get outside. And our expression should be one place. That's a squared. Plus wise hold square in two. We're differentiating with respect to wise, actually constant. So we'll be one. Sorry, will be excellently minus X Y into two. And two, uh, 00 So do why. Okay, something is fishy. One plus x squared, plus y squared will be here. One plus access choir. Plus why square then too. I will be okay. Do I buy one place? Access Square plus Y Squire raised the powerful. Okay. Okay. Here. It was some mistake. It should have been raised about four, so it should have been raised to the power. This was all right to risk wire by tourists to the power four. So it will be won by two. Similarly Yeah, it will be to rest in part four. So this will be minus one by four. I won by two. Sorry. Minus one by toe. Right. Okay. No. Well, you have this expression at our critical points, which is have why we're at minus one, comma zero. So this will be minus two one plus 12 minus two. Okay, this is minus two into, uh, zero. That's a minus two. This whole thing will be zero. So we'll be having one plus one to minus two minus two. Divide by one and 1 to 2 weeks to the par four. So we'll be having fall by 16. That is one by four. Okay, this is one before similarly, the value of this expression at our critical point. Why? Why one comma zero we have to plug in. So minus two will be as it is. This will be one plus one toe to Squire. Okay. Okay. Here. You should have been toe Squire, because this is quiet to a square. So four minus eight. So minus one by three, it should have been minus one by three. Because this is to to score four with eight so minus one by three. Similarly, here it will be too. And two is quiet into one. So similarly, it will be to square. The whole thing will be zero and divide by two days to the power four. So the saloon minus one by three. And here it will be extras minus one. Okay, So negative will be outside. Negative will be outside. So will be positive. Yes, we have got these values. Now we have to calculate one more thing, which is mhm but partial differentiation off this effects will with respect to why so we have to differentiate partially this quantity with respect to y. Okay, this quantity that is one minus X squared plus y squared, divided by one plus x squared plus y squared hold square or we can divide We can differentiate this quantity toe x y by one plus x squared plus y squared old square. So let us differentiate 1st. 1st only first month. This that is one of my insects. Square place where Squire by one plus x squared. Plus y Okay, one minus. Access choir plus fire squared, divided by one plus access choir plus y squared whole square with respect to y. So this will be one place access choir plus y squared one plus x squared plus y square. And this was Zito zero to why minus one minus X squared plus y square. We have, uh, kept this as it is indifference. We lifted and we will be differentiating this with respect to y so. To one plus x squared plus y square in two. Do what? Okay, this was whole square. Okay, Do I buy one place? Access choir plus y squared whole square. So value of this expression at minus one comma zero. That is first critical point will be C minus the this will be zero all become zero and we'll be having negative off one of minus 100 So everything will be zero zero and value off this the same expression with respect to why at one comma, zero will be This is the 1010 So this will also be zero. So both values are zero. Now we have we know that discriminate and at X come away in f xx f. Why? What minus? Fx, Where Whole square. So at critical point minus one comma zero Let us find the discriminate. The minus one comma zero D minus one comma zero is F XX at minus one comma zero. So this is one by two for F Y. Y is one by three. One by two into one by three minus zero That is one by six and we have f x x at miles one comma zero equal to one by two. So this is also greater than zero. This is also greater than zero. So both greater than zero which means the function have local minima local minima at, uh this critical point that is minus one comma zero. Now, if you talk about critical 0.1 comma zero one comma zero, that is second critical point D minus one Comma zero will be equal. Toe minus one by two. Minus one by three minus zero. That is one by six and F x x at minus one. Comma zero. Oh, sorry. One comma zero. Okay, it waas one comma zero minus one by two. This should be on commodity, not medicine. Comma zero minus one by two. So this is positive. This is negative. So local Maxima at one comma. Zero. Thank you so much.

In order to find the critical points of a function in two variables, we first need to find the partial derivatives with respect to each variable. So if I have a function f in X and y of X, why e to the negative x minus Y I'm first going to find the partial derivative with respect to X, which is why times e to the negative xnegative Why plus X y e to the negative Xnegative. Why times negative one and I can simplify that. Why? Times e to the negative x minus What minus why? Times one minus x. Okay, now the partial derivative with respect to why And that gives me X e to the negative x minus y plus x Why e to the negative x minus y times naked of one and Aiken Simplify that to x times e to the negative x minus y one minus. Why now? In order to find the critical point, either one of these partial derivatives has to have a point where it doesn't exist, which is not the case here, or they both have to equal zero. So I'm looking at setting the first partial derivative equal to zero and the 2nd 1 as well. Now e to a power can never equal zero. So I can ignore this piece because that's never going to give me a zero as a factor. So I'm looking at either why equals zero or X equals one and f y equals zero. That means the X will equal zero. And if x is one, then why is one So those are my two critical points 00 and 11 Now the question is what's happening at those points? Local minimums, local maximums, A saddle point. In order to find that out, I need to look at the discriminative. If you recall the discriminative. Is this function right here squared. So I need to find the second partial derivatives. So let me start with the second derivative with respect to X. Okay, how this is rather long, but not too bad. So what I have is what I have for my second derivative. Simplified is negative. Why times e to the negative x minus y times the quantity two minus x for the second derivative. With respect to why what I end up with when I simplify, it is negative. X times eat the negative X minus y times a quantity to minus y And I also need to know f sub x y And when I do that and simplify it, I get one minus x times e to the negative x minus Y times one minus y So what I'm gonna do to Dio is free to these points is to figure out the value of the discriminative. So I'm going to make a small table. I'm gonna show my two points. I'm gonna have the value of the discriminative at those points. I might need to know if the second partial derivative with respect to access positive or negatives I have a place there for that and the result. So from the 0.0 the value of the discriminative negative. So that tells me I have a saddle point at the critical 0.0.0 Now the 0.11 gives me a positive discriminative when I plugged the value in and evaluate. So I need to look at my second partial derivative with respect to X. That is negative, and that combination tells me that I have a local maximum


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