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Find all local oxtroma lor Iho function I(x,Y) = 4x" 9xy 9y7 13*Soloct tho comocl choico bolow and fll in any answer boxes within your choiceThoro aro local ma...

Question

Find all local oxtroma lor Iho function I(x,Y) = 4x" 9xy 9y7 13*Soloct tho comocl choico bolow and fll in any answer boxes within your choiceThoro aro local maxima localod al (Simplify your answors. Typo orderod pnirs. Uso comma (0 soparale answers a5 needod )Thoro are no local maxima

Find all local oxtroma lor Iho function I(x,Y) = 4x" 9xy 9y7 13* Soloct tho comocl choico bolow and fll in any answer boxes within your choice Thoro aro local maxima localod al (Simplify your answors. Typo orderod pnirs. Uso comma (0 soparale answers a5 needod ) Thoro are no local maxima



Answers

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=x^{3}+3 x y^{2}-15 x+y^{3}-15 y$$

In this question were given a function f x Y ICO Jew to expose three just two y about three minus nine X square because three y squared minus 12 y the first time we need to find the first with respect to the X So have the sixth X Square minus 18 x nico 20 Now we you can fact we can dividing everything by six and then we can factor the X outside. So when the X minus three equal to zero, Therefore we have the X equal to zero X equal to three. Now the f y, we get equal to the sixth quite square and then we have here will be plus six crime honest drown We call to zero dividing everything by six and began the y squared plus y managed to equal to zero Here, listen, even only if why plus two terms with the Y minus one that mean coaches also we have why you go two months to what you go to one. So we have the critical point here. It would be zero minus 201 It can be the three months to it can be the 31 Now the next time we need to find the second brush on the river too. F x x then we get equal to the 12 X minus 18. Similarly, the f y why get equal Thio the Here we have the drought y plus six and then the f X Y equal to zero. Now we will make a table 2000 the sign of each critical point. Now one to three on the farm here. So we have the f x x times f y Y minus f X y and this one with the f x x. So the first point we have to be zero minus two. Then we have the one we have the three months to. Then we have the 31 Now we can test it now. So we have an F and upon Germany. Still, we have this one will be the minus 18 terms with the Manus students and we have the minus 24 minus anything and now minus their square. Then we get this one will be positive here, So f x X equal to minus 18. So we smaller than zero. Therefore we have this one will be the local, Um maximum. And upon the one, we have manners 18 times. Schools that we were the one in Saigon. 18. When is it was square system smaller than zero. Therefore, we have the set upon here now. And upon three minutes to so 36 three times 26 minutes 18. You go to the 18 times with the minister inside and we got the minus 24 plus six equal June minus 18. And the manager squared is only with money. Danso. So have the set upon here as well. And another 31 We have the 18 times with one in 718 manager square cells and positive and we have the 18 here. It would be positive. Therefore, listen, we'll have the local minimum.

In this question was given a function. I am. Thanks. Quiet. Decode you for X qy minus expelled for minus wipe our farm And the first time we need to find the F X. Then we get go during the for wine. Oneness for export three, we said, is an equal to zero. Do we get a Y e? Go to the thanks. Power tree something. And we do the partial derivative, especially the y by Semitic When Shingen Uh, thanks. I'm Dan minus saw. Why about three there was sent on equal to zero. Their function. Get now ex will equal agenda. Why about three? And then it was booed This Y Yeah. Into the second equation. There was thinking the X We go to the expel a tree and in power tree. Then we shouldn't get include to the expound nine. So actually job about nine means that we should get the X minus expert on night. Go to zero. So x we fucked around. The one minus expel. Ain't you go to zero. So we have now ex very coaches. Uh, ex coach Blessing minus one Onda. We put the one minus one. Yeah, and they would put for the X, which is zero whyy coaching export to go big white coaches zero When we have let me right down to the three distinct cases here, girl on, then have while we go to one a swell There's a minus one. Then why we put u minus one and now we have totally three critical points Here we'll be 00 way Have 11 then we have minus one minus one. Now, the second thing which you do will be the second bash on the river too. Respect to the X And then we get equal to the minus 12 X square. Similarly, the F y why they were getting equal to the minus 12 y square and f x y we get ego to the fall. Now we can make a table to test that It's critical point now. One, two and three. Yeah, we have. This will be the f x x times f y y minus f X y square. Now this one will be the f Thanks. Thanks on. Now we have the critical 0.0 were 11 way have minus one minus one and the and zero which again does zero time zero, then minus four square. So it doesn't smaller than zero. Therefore we have this one will be the center point. Yeah, And now for the 2nd 0.11 we will have the minus three times minus thrill minus four square citizen be greater than zero and f x x. Here you go to the Manus thrill smaller than zero. Therefore, this point here we will have will be the loco Maximum. Now, for the third point we have, this one will be minus trail times The money trail on DTI 10 months for square. It could be greater than zero on before we will have this one f x X pill. Go to Minister A smaller than zero the phone and we're gonna have dot hoco maximum.

This problem. You're given these shown function some f of x y, and were asked to find all of the local minimums, maximums and several points of the function. Now, our first I've been doing this is going to be the to find the partial with respect to X and the partial with respect to why everyone a successful with people to zero on the pressure with respect to X is going to be to eggs. Plus why, plus three and again we're gonna set that equal to see a rope, the pressure with respect. So why is going to be X was to why minus three is equal to zero. Now we have to somehow solve for X and y. And so we're going to think of these as to different equations in the same system, and I'm going to solve one of them for why, and plug that into cells where X in the other. And so I'm gonna solve the 1st 1 for why, and I got the why is equal to negative two x minus the re, and so I'm going to plug it into the second equation. And when I do that, I see that X plus two times negative two X minus three and again minus three, I think. What? Zero. And when I distribute the Tuinei get negative. Four x nine a six minus three is equal to zero, you know, So I can see that I get negative. Three X and then this is negative. Six minus two. Your negative. Nice. I'm gonna add 90 both sides. I get negative. Three X equals nine. So x equal to negative three. And I'm going to plug that in up here I get Why? So why is equal to negative two times negative three minus three. So why is equal to negative two times negative three or six minus three, which is three. So I found my critical point. I only have one in this problem. Negative three, comma three. And now I'm going to use my info appear to find. But there's a local minimum. Excellent. More settled. Got to do this. I need the partial with respect to x. Twice the party with affects their sector. Lie twice in the partial respect connects them with their sector. Why so I partial with her stuck to x twice. When I look at this to find that And that is going to be too now, my personal, with respect to why twice. But I like this one and that is actually also going to be, too. And then my pressure with respect to X and then with respect. So why is going to be one now up here? As you can see, what we're gonna do is take this multiply times this and subtract half of the partial with respect to X. And I would expect a why squared from that someone like two times two and subtract one squared from that. And so I got four minus one, which is equal to three now. If you had an X or Y, you would be able to plug in your ex and my values to see what happened when we club those extra my values in. But here, as you can see, we just have a constant and so that numbers were always want to get as we take precautions. Expected X twice multiple. It sends a partial check the wide twice and subtract the pressure. They're stuck to accident with your sector. Why? And as you can see, that is greater than zero so either have a local maximum or a local minimum. And we have to look at whether the pressure with respect to X twice is greater than or less than zero. As you can see from here, it is greater than zero. And so we have found that we have a local minimum. So local men, what we want to do it's fine f of negative three comma three to find with that value is and so I'm gonna plug that in up here. So we have X squared plus x y. I was gonna write the equation again down here, plus y squared was three x minus three will impose for if we plug in negative three and three, we end up getting negative three square, which is nine plus negative. Three times three, which is negative. Nine plus y squared, which is nine. His three great is nine post three times eggs, three times negative. Three is negative nine and then we got a minus three times. Why? Why is three tonight of three times three is negative nine and then we get plus for as you can see, this nine in this negative cancel out cause equals zero insane for this form and this one. And so we get negative nine plus four and then is equal to negative thought. And so we have a local minimum. When we plug in excess equals negative three. And why is equal to three and at that point asked of negative three three dysfunction value is negative, but

In this question were given a function. F x Y ico Thio Able X Square, plus Boy Square minus four X and the first time we need to find a first crush. Larry with goes back to the eggs. They let me go to the EBA X Square plus y squared minus four x by the general. We need to Thames that you x minus, for we send it on equal to zero. And then it will be even only if that your bank minus form was equal to zero even only if thanks musical to two. And now the passion, the river. Those particular why on the go to the a X Square plus boys Grandmothers for X, then by the general need two times two y recently go to zero should even only if the two y equal to zero, even only if one must equal to zero. So we have one critical point on Lee Now. The second step, we need to find a second brushing the river to Don't get Nico June uh, a X squared plus boy square minus four x we for the county. Then we will have it will go to the two X minus four square and then minus will be plus here then too and then Similarly, if we have the f y y, you will equal Thio the A X squared plus y squared minus four x on Now we fucked it out. Then we'll have equal to the to for White Square And then way We're, uh plus that you on Dan. We will have the f x y We should get equal with you that Ava X squared plus y squared minus four x And we will have doesn't with a two x minus four times with the to why And now we consider at the critical 0.20 We will have the f x x times f y y minus f x y square here This one here and we put the chooser inside. We have the far minus. It will be a power. This one will be on with positive here on Then we put the two found inside chooser inside we have the to. This one will be zero Now on, then we have a two. So we have this one will be that a power off the we have here would be we confront that outside. So for minus I eat so able minus four outside and inside we have the two and then times with the we have and is only we have times two and then minus the evil minus four. We still have it here and then we have on the Jewish There will be This one will be zero now, so everything will be zero. Uh, this will be managed zero square and therefore we should get equal to the positive. Here it would be positive, whereas under F x X, it will be bonded as well because they were looking to the f x x and we're constitutive that far we have this point will be in the local minimum at a 0.20 and f and the 20 to equal to the evil minus four.


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