5

Find the Iocation of the minimum absolute extremum for Ihe lunction...

Question

Find the Iocation of the minimum absolute extremum for Ihe lunction

Find the Iocation of the minimum absolute extremum for Ihe lunction



Answers

Find the relative maximum and minimum values. $$f(x, y)=x^{2}+x y+y^{2}-y$$

Talk about question number five. So first with the friendship with respect to eggs. So we have three extra square minus three wire taking wires a concert. You find a wire that is differentiating with respect of wires. So we have three wire square minus three eggs treating excess of concert. We create both to zero. So we have three X squared, minus three wise equated to zero. This means that extra squares while on from here we have three wild square minus three x is equated to zero. This means that while square as eggs so from here we please the value of y. So this fathers and device toe x square who square as eggs. This means that X rays to the poor Force X. This means that ecstasy the zero or X 01 f x zero. Then why is also zero and effects was one. Why is also one probable? So we have two points of interest here. 100 on 11 Now let's find f x x. So we have f x X s differentiating this again with respect to X. So we have six x f. Why? Why would be differentiating of Why with respect the wires. So we have six wives if you have f X wires Differentiating this with respect to wise. So we have negative three so f g will be f x x times y y minus f x y square. So this consult us 36 x Y minus nine. If you place X and Y zero, we have DS negative. So there won't be any threat of extreme er if you place X and Y as one and one that we have d s positive because it was 36 minus nine, which is 27. So it's positive. And since f x x at this point 66 so that will be six. So since ethics is also positive, it means that we have a relative bonoma and the value of the little minimum would be, uh he plays excess one and why it's one. So we have one plus one minus three, which is minus one. So this is the value of relative minimum on the point at which it occurs is one comma, one

Talk about question two. We need to find a little Maxima and minima off this particular function. So let's first differentiate this with respect to X. Treating y is a constant, so this will be two X over here. We get by on the other terms. We just busy over the vice featured as a concert. Let's to appreciate this Petra speak to Wise, so it should be treated as a concert. So we'll get X Plus two y minus five. Now, in orderto find our points off interest, we equate effects to zero. It means start to express by is equated to zero Mr Wise minus two works and like ways, we have explosives too high minus five is equated to zero as also we have expressed to her minus 50 for the value of y as minus two weeks. In this equations we have minus two X minus five is equal to zero. Things gives us negative. Three X is equal to five. It just gives us the value of excess negative five or three s. So this is the value of X and this value of X the value of why will be negative food negative five or three. So that comes out as 10 or three. So a point off interest would be negative. 5/3 and 10/3. Now we have to find whether this has a relative maximum minimum. So will differentiate FX once again to get f x X. That is the derivative off FX. With respect to X. Once again, we'll just get to hear, because why is treated as a constant like ways we find f y y so every y y from here, treating extras a constant. But again we do and then we find f x y so differentiating FX with respect, why will give us just one? Uh, Now we'll see. The sign will find d d s f f x x times f y y minus f x y square. So this is nothing but four minus one, which is three. So we noticed that he is positive on F X X is also positive. It means that we have ah, relative minima at a two point of interest at a critical number and that minimal relative minimum would be so. The value off f X Y is point would be, please excess negative five or three. So we have negative five or three. Hold square, plus negative 5/3 times. Why? Why? Sten? Over three plus wise, where we just stand What? The whole square minus five times Why? Which is again? 10 or three. So this comes out of 25 19 When is 51 9 100 over nine when 50 over three. So thats is actually nothing, but let's clear it up So we have. If you feel going to simplify this, this is 100 minus fifties 59 15, 50 and 50 plus 25 years 75. So we have 75 or nine minus 50/3 Which one simplified comes out as negative 75 or nine which is nothing but negative. 25 or three. So this is the relative minimum at the point, minus five or three comma Turnover three

Let's talk about Question three s in order to find a little maximum minimum force will differentiate this with respect. Works keeping y as a constant. So we have two by minus three years a square, and then we'll do the same thing. Keeping fixes a constant So we have to hire, uh, minus Dubai over here once again. Actually, it will be two X because X is taken as a conference. The differentiation of why, with prospective violence just one. So we have two x managed to weigh. Now we're here. We quit affects 20 So we have two y minus three extra square is nothing but zero. So from here, we have to Wire is as three. It's a square scholar equation one and like ways. From here we have two x minus tree Where square zero. The value off why from this equation is nothing but three x square or two. So this is equal to zero. If you simplify this, we have four X minus nine. X square is equal to zero. This means that exist. Taken out, we have four minus nine x as equal to zero on this one correction Here This is actually to wise. So we have to rewrite this way mystically Go three Why we're here. So let's rewrite this. As we have two weeks minus two y is equal to zero. So from here we'll place the value off two wires. Three X squared is equal to zero. Access taken out. We have two minus three x 0 to 0. This means that either x zero or two minus three x zero to ministry 80 than from here We have three Xs Stewart me start access to over three s So we have two values off extra. Me I we have values of X from here on the corresponding values of why would be effects zero on two y zero, which means that y zero if X is to over three onto A s three times for over nine. It means that why else on we simplify this, we again get two or three. So there are two points off interest. 1001 is two or three to over to you. All right, let's find out whether this has a little maximum minimum or or neither. So if you find fxx f x X is nothing but the derivative off FX respect the works again, which is nothing but minus six x. You find out why Why f y y is again the derivative with respect. Why? Which is nothing but minus two on we find f x y, which is the relative of FX with respect. So why so with respect to why and features to be to Then we find f we find d which is f x x times f y y minus effects while square. So this consult as elects minus four. Uh, Now, if you place our first value in which x zero then d is actually minus four. So whenever D is less than zero, there is no, uh, minimum or maximum So we cannot really take it on if access to over three then we're just interested in the science. Over three minutes it would be 12 minus four, which is eight. So it since D is positive it will have a rid of Maxima or MINIMA. And we also see that if access to over three have, xx will be negative. It means that we have a relative since f x X at this point would be negative So we have Ah, relative Maxima. And that relative maximum value would be FX that effects FX common. Why would be with taste the values of X and wires. So we have two times. Ex wife, both hard. Global tree minus two will treat you with nothing but a photo. 8/27 minus. Here we have for over nine. So this comes orders 8/9 minus 8/27 minus 4/9. Took common denominator. 27 we have 24 minus eight minus. Uh, yeah, this will be, uh, 12. So we have to inform. Minister is 12 and 12. Minus eight is four. So we have four or 27. So this is the value of the relative Maxima wild. This is the point where the relative maximum occurs.

Talk about questions. Well, it's supposed to find relative X females years so false will differentiate with respect to eggs. Uh huh. We have two weeks, three fires. A concert will differentiate with respect to y So we get negative two y we differentiate with respect to X again. So we have to differentiate with respect to buy again. So we have negative toe on f X y would be cereal as there is no y here. So let's equate effects on a vibe each 20 to get the critical point. So we have two weeks and negative. So why Fort equated to zero? This means that excuse you and why is also zero. So the critical 00.0 let's find Dino so D is f x x times f x y, which is two times minus two, is minus four minus FX wire square, which is just you. So we do. We're not supposed to do any further calculations. Clearly, India's negative. It means that it has no relative extreme ass


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