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3_ [Total 15 marks] Consider the functionf(,y) = 2 = e fv(a) [3 marks] Find the location in the (€,y_ plane of the critical point(s) of f (x,y), and give the ...

Question

3_ [Total 15 marks] Consider the functionf(,y) = 2 = e fv(a) [3 marks] Find the location in the (€,y_ plane of the critical point(s) of f (x,y), and give the corresponding value(s) of z. (Only consider points with finite values the coordinates, i.e. don't consider the behaviour of f at infinity). (b) [2 marks] Write down the equation of the tangent plane(s) to f (€,y) at the critical point(s) identified in part (a) , giving a brief justification of your answer: (c) [5 marks] Using the m

3_ [Total 15 marks] Consider the function f(,y) = 2 = e fv (a) [3 marks] Find the location in the (€,y_ plane of the critical point(s) of f (x,y), and give the corresponding value(s) of z. (Only consider points with finite values the coordinates, i.e. don't consider the behaviour of f at infinity). (b) [2 marks] Write down the equation of the tangent plane(s) to f (€,y) at the critical point(s) identified in part (a) , giving a brief justification of your answer: (c) [5 marks] Using the method of Lagrange multipliers, show that the extremum of f(x,y) subject to the constraint T + 2y = 4 occurs at the point (2, 1). (Again, only consider points with finite values the coordinates). (d) [5 marks] Evaluate the Taylor series expansion of f (€, y) about the point (2,1), keeping terms up to and including second partial derivatives of f _



Answers

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s). Find the critical points of $f(x, y, z)=3 x^{2}+4 y^{2}+2 z^{2}$ such that $-x+2 y-3 z=21$ and $2 x-3 y+4 z=15$

Everyone directions that we have in this example, Or if thinks I'm going, you quit X squared Those two words square with the mainstream functional G off X roi equal four x minus six Oil because 24th together around immigration forms. Responsive. If my most longer G it was zero, this will mean partial differentiation. Off X will be equipped with a bunch of foundation off X, and the first concert is that for excellent boy will be to Ace Mine's London abortion, but by the person from Station Off fixes the second function, which is four. This will give us two X equal for long, full of the same concept. Departure from fishing before I will be able. Full boy brainless longer nearly six. This should give us four Loy because negative six lot is our first equals for our seven requirement. If we take X, people presume this will mean that longer people zero. Which means that worrying before you embrace verse. That's that's mean. If if X is equal to zero or before this court to resume their long live will be because you ends Expo on the corresponding expert work with the old speak one is our second for ourselves permanent if you took If we take the two functions which is two X full longer and for warring equal negative six long in front of different The first equation by to worry This will give us for X roi equal eight. War longed up and we want to. But second equation Boy X four x warning The equal negative six x long by taking unequal for the two equations in boy Lunda were with *** to six x longer and this will give us for one equal new to ST X Way Will takes tourism and implement is ah Jeanne fix on droid. We would have four X minus six. Deploy negative street over four X Peak 1 25 This will give us an X equals two 25 over a On was there so we would have you, Lloyd equal. Negative. 74th over 34. This is our critical point to see if, uh, this critical boy is corresponding to a minimum or a maximum value of f. We will apply the slow. Really? Which leaves that. Are this club off the? Is there slow? 00 G fix genetics. Enjoy function one over X, which is the equivalent negative three or four you see over full. This means if we moved a distance off X, we will move in distance before new. Expect four on by going Negative street. This way loses dysfunction tohave another point on the line running away from our critical point which would be our X to x two equal 649 and going to equal negative voice. I want to also can have an excess tree equal then for mine. And Boy Street equalled military in Windows by implementing these two values off X and boy into the function f x If x ongoing, we we have and if one equals two 18.4 if two equals to do if she because Because I'm sorry, if two will be equal 102 want some dissonance f street with the bull 254 foreign Dismiss. This means that by running away from the court from the point of the critical point that we got from the ground integration, a differentiation we way is the value of his F function increase which require that they be is corresponding to a minimum value off. If and this is our answers, thank you

Here. We're trying to find critical points of F equals x squared plus y squared plus z squared. So basically a bunch of nested spheres. So if f week was a constant constant greater than zero, get a sphere. And let's see here, then we have two constraints. And these are two planes again and these two planes were intersect in a line. And then that line is what we're trying to optimize this function over. So optimizing this function subject to the line, that is the intersection of these two planes is the geometrical interpretation. So we have G equals F plus lambda, one, C one plus lambda to see two. So again we have five variables here. So we take the derivative respect to each one of them, X Y, Z, I am the one latitude And set those all equal to zero. You get five equations and five unknowns. But there are linear equations. So we have one solution and you can find them by back substituting or using matrix algebra. Or lots of different ways of finding those. I didn't go through all the ugliness of that. But what you wind up with is X equals minus one third, Y equals minus two thirds, Z equals five thirds. And then these are two lambdas here plugging this back in and we get F equals um 13th. So the Uh the optimal solution would be on a sphere of radius squared 10/3. And we can also basically what I did here to get this plot. As I took these two equations, solve them for X and Y. In terms of Z plug those into here's and then I get a function of Z. So F bar is just a function of Z, and then I plotted it and you can see here that you get this minimum here And it occurs around 10/3 And happens occurs for z equals 5/3 right around here. And then if we found that we could then use their solutions, we got from up here of X and Y in terms of Z to find X and Y. So again, this is consistent with basically solving our constraint equations and then substituting back into our function To eliminate two other variables. So again, it's good to check this kind of to make sure that you're consistent.

So in this problem we are trying to find the minimum points. In a matter of a function, A T equals to the square root off X plus two square plus why minus five square plus C minus one square. So this is a distance between the point X Y Z and the point minus 25 and one. But we don't have to use this function. We can. In fact, we have. We have to use the square off this the square of this distance function which, uh, it's very easy to compute this decorative. Thank you, e. In fact, you see if we can minimize this function. If that means we can also minimize the function. D eso the constraint questions G equals to two x plus three wide plus 60 minus 10 equals zero. Um so you slap around much method brilliant off every equals to two X plus four on tour Y minus 10 and the Z minus two and the breaking off G equals 2 to 3 and six so that are on. The prime method gives us a system of equation. Toe X plus four equals to two. Lambda two y minus 10 equals the three Lambda to Z minus two. We close to six laughter. Then we can re express X y Z in terms off Ladha. So, for example, X equals to love the minus two. Why it cost three lambda last developed by two the equals 23 lamp That plus one. Then we use this, uh result plugging back to the constraint equation G close to zero. That gives us two times Lambda minus two plus three times three lambda plus 10 half. Um, but the process six times three left that plus one equals 2 10 Mhm. So we have 24.5. Lambda equals two minus seven. I think. Therefore, Lambda, because to minus 2/7 and they once we solve, uh, Lambda we have X equals two minus 16. Divide by seven. Why close the 32/7 z equals to 1/7. So the minimum point off the function happens when x e x y z equals to these values. That means when X y Z equals to these values. We have the smallest distance, so the distance the minimum distance equals toe brutal x plus two square plus y minus five square plus C minus one squared. We just plugging these values into X y z A. That the result will be just one that's

The problem is very, very similar to the previous problem but now we're finding my critical points of X plus Y plus C. And now our constraint is a sphere of radius six. So we form our augmented function with the Lagrange multiplier. Take partials on satyam equals zero. So we get one plus two lambda X equals 01 plus two lambda Y equals 01 plus two lambda, Z equals zero and then C equals zero. So she is here in that equals zero. And we can then uh we then you know just back substitute and you know eliminate variables. And we finally find out that we had to solutions one is X. Y and Z are all two times the square to three. And lambda winds up being minus 1/4 times the square three. The other solution is X, Y and Z are all minus two times the square to three. And Lander winds up being 1/4 times the square three. So this plug that into here and we obviously get six times square to three. This gives us -6 times the square to three. So this is a maximum, this is a minimum. And if we if we um What I did is I solved this is for Z got you know the 22 solutions e equals group Z equals plus or minus something, plug that into here. So I get two branches of F and I plotted those two branches and again we get this kind of like Mylar but inflated balloon like structure here and it shows that again we can see where are, you know, we have a maximum over here and the minimum, basically it's underneath there. And those are, you know, we can see that they're on this kind of diagonal, um, that sequence by cuisine.


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