2. Let f(c) = 325 513 _ 5 for -3 < € < 4 Find the ordered pairs (€,y) corresponding to the absolute maximum and absolute minimum of this function_...

Find the absolute maximum and minimum values of the following functions on the given set $R$. $$f(x, y)=2 x^{2}+y^{2} ; R=\left\{(x, y): x^{2}+y^{2} \leq 16\right\}$$

To find an absolute minimum or maximum value for a function in an open or unbounded set. We're gonna look at two places. First, we're gonna look at critical points of our function, and then we're gonna examine what's happening with the function as we near the boundary or as we get farther and farther and farther out on our set. So first we have a function in X and y, in this case or function is X Plus three y. In order to find the critical points for this function, we first have to find the partial derivative with respect to X and Y partial derivative with respect to X is one and for why is three. There is no way to make either one or three equal to zero, so there are no critical points within this function that we can look at. So what's happening at the boundary? Well, for this problem, the boundary says the absolute value of X is less than or equal to one, and the absolute value of why is less than or equal to two. As we look at this, the minimum value for the function would happen when X and y are both a smallest possible and we'll have the maximum value and X and wire. Botha's biggest possible, however, these are not. We'll never get there will never have the biggest possible value for X because we can always get just a little bit closer. Toe one will never have the smallest value of X because we can always get just a little bit closer to negative one. So there is no absolute minimum and no absolute maximum for this function. No critical points and nothing at the boundaries.

In order to find the absolute minimum and maximum values of a function, we need to look at two places. These minimum and maximum values were gonna happen either at a critical point or at some place on the boundary of the set. So, first, let's find the critical point for this function. To do that, we need to find the partial derivative of that function with respect to each variable. So first partial derivative with respect to X is for X minus four and the partial derivative with respect to why is six why? So where does this equal zero? Well, this is going to equal zero when X is one, and why is zero? So that is the critical value for this function. But what's happening at the boundary? Well, in order to represent that the simplest way possible, let's put this in terms of a parametric function. So we're gonna do a substitution. I'm going to say that X is co sign of fada plus one, and why equal sign of data. And if I substitute those back into my original function, that will give me a function. Gonna call that G since I'm changing the variable I'm going to change the name of my function and that gives me too. Times Co sign Fada plus one squared minus four times co sign of Fada plus three times thought. Sign of fate is squared plus two. And if I simplify that, what I will end up with my for my function G is to times co sine squared of Fada plus three times sine squared of data. Now, to find my critical points along that boundary, I just have to take the derivative of that function G. So the derivative of my function is four times co sign of Fada times negative, but that outfront sign of data plus six sign of fada co sign Seita or to simplify that the derivative is too Times co sign Seita sign of data. So where is that equal to zero? Well, either co sign of Fada equals zero or sign of fate of equal zero. This is gonna happen at two places for co sign to be zero equal. Either faded equals pi over two. Ortho equals three pi over two and that gives me an X and a Y For each of these points. If I substitute that data back in for the values of X and y that I picked earlier. That gives me X equals one and y equals one. Our X equals one and y equals negative one. No, over here for sign of fada to be zero. Either say two equals zero or say that equals pi and again substituting those back in to find the corresponding excess and wise. That gives me X equals two and y equals zero or X equals zero and why equals zero? So that is five points. I need to check one critical point within the function and four points on the boundaries. So to find out the value of the maximum and minimum for the entire function, I'm just going to substitute each of these five pairs of X's and y's back into the original function. So the value at one comma zero the function has a value of zero. At the 0.11 the function has a value of three and likewise one negative one. Give me a value of three two and zero gives me a value of two and 00 gives me a value of two. So the answer to my question The absolute minimum gives me a value of zero ends. My absolute maximum is a value of three, and that happens twice along the perimeter.

In order to find the absolute maximum and minimum values of a function, we need to find two things First, any critical points for my function and second, maximum and minimum points at the boundary of my set. So let me start. I've been given a function. Let me start by taking the partial derivatives with respect to X and Y. This will help us find our critical points for our function. So the partial derivative with respect to X is two X the partial derivative with respect to why is, too why minus two. Now, where do those equal zero? Well, I can see from the first equation the X would have two equals zero. And the second equation tells me that why would have to equal one. So that is the critical point for this function. Now, though, I also need to look at the boundaries. And I've been told that my set is all numbers X and y such that X squared plus y squared is less center equal to four. Now that's a circle with a radius of four or less. So the easiest way to see what's happening at the boundary is to set this in terms of a parametric function, I'm going to make a substitution. I'm going to say that X equals to co sign of Fada and why equals to sign of Fada and I'll plug those back into my original function that I was given. Since I'm changing my variable, though I'm going to make I'm gonna call this function G of Fada and that's going to be four co sine squared of data plus four signed square to state up minus two times Why? Which is to sign of Fada plus one. Now, if I simplify this a little bit, this piece right here equals four sine squared state of plus coastlines. Court data equals one, so that equals four and my whole function turns out to be five minus five minus four. Sign of Fada. So my maximum and minimum points along this function happened when it's derivative equal zero. This is a function with only one variable, which makes finding my maximum minimums much simpler. So I take the derivative of this function. I get negative four co sign of Fada. Now when is negative? Four coastline of Fada equal to zero. Well, when co sign of fainting equals zero, and that gives me two possibilities. Either Fada equals Pi over two or faded equals three pi over two. Pi over two gives me I'm gonna write this over here I have X equals zero, and why equals two. And if a T equals three pi over two, I get X equals zero. And why equals negative too? So two points. I need to look at the boundary and one critical point. So I'm just going to make myself a table. I'm gonna have my points in one column and I want the value of the function. If I do this out 01 If I plug that into my original function up here, that gives me a value of zero. The 0.2 gives me a value of one, and the 10.0 negative two gives me the value of nine. So, as you can see, this is my absolute minimum zero. And with a value of nine, this is my absolute maximum of my function.

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