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Find the local maximum and minimum values and saddle point(s) ofthe function. If you have three-dimensional graphing software,graph the function with a domain and v...

Question

Find the local maximum and minimum values and saddle point(s) ofthe function. If you have three-dimensional graphing software,graph the function with a domain and viewpoint that reveal all theimportant aspects of the function. (Enter your answers as acomma-separated list. If an answer does not exist, enter DNE.)f(x, y)= 4ex cos(y)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x, y) = 4ex cos(y)



Answers

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.
$f(x, y)=x^{2}+y^{4}+2 x y$

The problem is finding the local maximum and minimal values and set a point over the function. If you have three dimensional graphic software, grab for the function of it at the monument point that reveal out important aspect of option. First we compute bath. Relax. This is the cultural two ax Linus X Square plus y square hands you two next to fax. After a while, it seemed much, too. Why How's into naked backs? So like fossil? Hm? You caught you zero we have. Why is he called? Zero X is equal to zero War two. Then we compute. So we have two points to a critical points. Zero zero and two zero, then the computer. Second, partial derivatives. Five sacks like Sly and what I want a fax X is equal to you. Shoo minus or axe plus X squared. Plus y squire. How's need to negative flux? A fax. Why is he going to negative, too? How's into making bucks? Why why is he going to to hamasleader connect your Why so I'm disappointed. Zero zero. We have our fax snacks. Is he going to Why? Why does he want to on a flex? Why, it's equal to zero. The war it's greeted at zero since FXX is created at zero. Zero zero is a local minimum as a point two zero, we have a lack sex. Is that a cartoon connective to house into Conectiv too? And why? Why? Yes, they caught you, Cyril. It is why? Why is he wanted to now flax? Why is he going to zero? So the is equal to negative. Ford has inter. Negative too is less and zero at this point is a saddle point. Now let's look at is a graph for off the function from the graph where we can see at this appointed zero zero to function as a local minimum. But that's the point zero. It is neither a local minimum, no local maximum, so it is set a point.

Okay, so we're trying to find a local maximum in for dysfunction are here. So I'm going to do that first thing you do the first derivative in both the X and y so f of X is gonna be equal to two X plus a lie. There's gonna be equals zero. Why is gonna be equal to X plus two y plus one that's also gonna be equal to zero. So because he here that two X has been really cool to negative wire and it also just say that I get to exiting the equal. So why? So why don't we put that into this equation over here? So we have X plus two times two x plus y equals zero. So x minus court excellently was only one maybe three accident to negative one exit do even with 2 1/3 And if X is equal to 1/3 that means that's why we're here is gonna be equal to now You two terms one cruise, which is it can be equal to you two birds. So that means our point is gonna be 1/3. There were 2/3. Now I need to find out if that point is gonna be a maximum and the minimum maximal or minimal. So let's see, we we take the second derivative in respect to X of our first of in District X, and we're going to get to and why? Why? Is gonna be too as well of x y is gonna be equal, Teoh One So d is going to be go to f expects of our point science effort. Why? Why? Minus f of X y square sort of d is gonna be with the two times two minus one square. He's gonna be equal to four minus one secret of three created in zero and then also from X X is you could go to, which is greater than zero. Therefore, we have, ah, minimal A local mill, uh, 1/3 get 2/3

The problem is finding the local maximum and minimum Williams and saddle point of a function If you have three dimensional graph in software. Ralph the function whether the boy and we pointed that revealed out important as backs off the function. So first computer, half tracks. This is control. Why us, Axl? I times negative x times Each is a power connective x squared plus y squared to successfully asik Untrue. Why harms one minus X coyer times each of the power of negative I squared plus y squared over two I shall have wine a competition This is equal to X times one minus y square house needs connective exploiter a slice coyer over to you Latin Bolshevism, you called zero We have X It's the culture plus minus one. Why, you country zero war Why you could to plus or minus one I want to zero on behalf foul points plus or minus one Awesome minus one plus my swan minus plus one and zero zero. Then we computed half second Partial derivatives have x x This is the culture negative. X y hams three minus exc Lawyer How's e to the power of elective X square plus y square over to you. Why? Why This is equal to negative attacks by Pam's three Mass Life Square. How sweet is that? How often effective, exploitive plus supply squire over to you on DH X y It's in country one minus tax Coyer Times one minus y square. Sleep it off Negative exploiter. That's why squire over too. So that's a point one one. What's the point in next two one Next one we have fax. Fax. It's the Contra negative two times each with a power off ninety one. Why? Why, if they caught you negative two times each with a part ofthe next one. I have x Y you seek Or to Cyril. So half the court. You or Ham's Inter Negative, too. This is acquitted and zero things off. X X is less than zero This two points Uh, no con maximum as it's appoint one next to one, and there's a point next. One one off Ex LAX If they go to two times the next one. One wife. His record two times to ninety one on a fax. Why see what? Zero. So is a torture or Tom City to meet you, too It's great, isn't zero since half XX is Colton and zero. So there's two points a local minimal. That is a point zero zero. We have half XX. It's equal to after. Why? Why? It's equal to zero. Ex ally. It's the culture one. So is that going to make you one last time? Zero. So it is a saddle point. Now let's look at this. A graph for the function so we can see that is a point to one one next one next one. It is a local maximum and that is point ninety one one one two one. They are local minimum, but that is a point of zero Cyril. It is neither local, maximum nor local minimal, so it is a subtle point.

Yeah. Here we are asked to find the local maximum minimum and saddle points of the function F. Of X Y equals X Y. E. To the negative quantity X squared plus Y squared over to. And so first we have to find the critical points of this function. So we will take the X partial derivative and the y partial derivative. Um Here this is a product rule. And so we will first take the derivative of X and then multiply by Y. E. To the negative X squared plus y squared or two. Then we will add X. Y times the derivative of the second term here, which will just be the term itself because it's an exponential E. To the negative X squared plus Y squared over two times the denominator of or the derivative of the exponent which is negative X. The two from the X where then the over to cancel. And we also have F. Y is going to be very similar. It will be X. E. To the negative X squared plus Y squared over to plus X. Y. E. To the negative X squared plus Y squared over to times this time a negative Y. So we can simplify these a bit. Can pull out A Y. E. To the negative X squared plus Y squared over two from each of them. Or the Y from the top one and an X times the exponent for the bottom one. Perfect. And then we are left with on top one minus X squared and on the bottom one minus y squared. Yeah. So if we rewrite this down here, we know that for a critical point the partial derivatives are zero. And so we have zero equals Y. E. With the excrement one plus X one minus X. And then zero equals X. E. With the exponent one plus Y one minus wife. And so we see for this top one here we have a couple of options. We can either make X equal to positive or negative one. Yeah or this exponential is always going to be non zero. So if we make why equal to zero out front? And that also works for the the bottom one we have Y is plus or minus one and X equal to zero as well. Yeah. And so this is going to give us a collection of points given by 00 zero. Or that is that is the only one with any zeros and the others are on one on negative one negative 11 and negative one negative one. So next we have to apply the second derivative test to each of these critical points. So we will take our second derivatives starting with the second x derivative. So here this is again going to be product rule. First take the derivative of the exponential. Mhm. We already know that we're going to pull down a negative X. So I will just write negative X, Y E to the negative squared plus Y squared over two times one minus X squared. And then now we leave the exponential the same and the derivative of one minus X squared is negative two X mm R f y y is going to again look very similar just swapping the X's and wise basically. Mhm. Yeah. Mhm. Yeah and for the mixed partial. Um So for instance the y partial fx this one minus X squared. There is a constant that gets added on and we have a product rule again. Mhm. Yeah. So we will just right uh first taking the derivative of Y. That leaves us with E. To the negative X squared plus Y squared over two times one minus X squared. And then now leaving the wine taking the derivative of the exponential with respect to why is going to give us a negative to why? So we will have a negative two Y squared E. To the negative squared plus Y squared over two times one minus X squared. Yeah. And so now we have expressions for each of these. We can simplify them but it's not super important since we're just going to be plugging in some pretty basic quantities. Yeah. 11 negative one in the 11 negative one negative. And then what we need to know about these is we need to know U. Values Fxx F. Why? Why the mixed partial F. X. Y. Okay. The value of the discriminate which as a reminder is fxx F X Y. Or I'm sorry Fxx F Y. Y. This F X. Y squared. And then finally based on those values we can conclude using the secondary of the test. Yeah. So plugging 00 into this. We see the first term in the second term goes to zero for fxx for F Y. Y. Yeah and for F X Y. We have E to the zero times one. So we actually have a one. Yeah for 11 we have zero in the first term and negative two the exponential will just give E to the negative one. So we have negative two over E. Yeah here for F Y. Y. When plugging 11 this first term again goes to zero. The second term we again have negative two over E. And for the next partial each of the terms goes to zero. And that will be the case for all of the um all of these other points the mixed martial will be zero because we have this one minus X squared on each of the terms. So now plugging in one negative one, the F. X. X. And first term will be zero. So in all of these cases were only looking at the second term here um and this negative will cancel basically because we have the Y. There and so this will be positive to over E. And the same will be said for F. Y. Y. Positive to over E. Again when the signs are mixed here this is positive two over E. And then when the signs go back to being the same, we retain the negative we have negative two over E. So now here we can rate the discriminates 00 is going to be a discriminative of negative one. Mhm. And then for all of the others we are going to have positive discriminates of four over East squared because our X. X and Y. I always have the same sign. And so we can conclude this first point is a saddle point because it is always whenever we have a negative discriminate it's always a saddle. For the others we have to look at the sign of fxx or F Y. Y. If it's negative, that means the function is decreasing around the point. And so it's a max. So for this first point we have a max and for this last point we have a max and then all the others are mins because the second derivatives are positive and so the function is increasing around them. So based on this we can conclude that there is a saddle point at 00 We have local maxes at the points 11 and negative one negative one. And we have local men's at the points uh one negative one and negative 11 Yeah


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