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Find an absolute maximum and minimum values of the following function on the given set R;f (y) = Vox? +9y 28x+ 28R is the closed half disk {(x,y)/x? +y <4with y2...

Question

Find an absolute maximum and minimum values of the following function on the given set R;f (y) = Vox? +9y 28x+ 28R is the closed half disk {(x,y)/x? +y <4with y20}

Find an absolute maximum and minimum values of the following function on the given set R; f (y) = Vox? +9y 28x+ 28 R is the closed half disk {(x,y)/x? +y <4with y20}



Answers

Find the absolute maximum and minimum values of the following functions on the given set $R$. $f(x, y)=\sqrt{x^{2}+y^{2}-2 x+2} ; R$ is the closed half disk $\left\{(x, y): x^{2}+y^{2} \leq 4 \text { with } y \geq 0\right\}$

But this problem, we want to refer to the graph and we're specifically going to be looking over the interval From 2 to 8 Uh huh. The first we want to find the absolute minimum electricity in this case. Um it starts off at to where it's equal to eight. Um but it goes down further and it reaches a minimum value of five. So we're going to say that the absolute minimum every good five and then the absolute maximum mhm. Um Since this is going to eight, we see that when it reaches eight the highest it'll get is not very high, so the absolute maximum is actually going to be at three x equals three. Um So when X equals three, we end up getting nine, so nine is going to be an absolute max.

In order to find the absolute minimum and maximum values of a function, we need to look in one of two places. It will either be at a critical point of my function or it was going to happen somewhere along the boundary of my function. So for the given function F, let's start by finding the critical points. To do that, we need to find the partial derivative with respect to X and with respect to why so, x First, that partial derivative is going to be four X and the partial derivative with respect to why is, too why, as you can see, what value makes us equal to zero the 0.0 So that is the critical point for this function. I also need to look at what's happening at the boundary to do that. The best thing to do is to recast the problem. Parametric Lee. I'm going to do a substitution. I'm going to say that X equals four co sign of Fada and why equals four sign of data and that comes from my boundary equation. X squared plus y squared is less than or equal to 16 which is a circle with a radius of four or less. So if I make those substitution, I'm going to put them back into my original equation. But since I'm re casting it in terms of Seita, I'm going to give it a new name. GF seda, and that's going to become two times four co signed status squared, which is 32 co sign of data squared. Plus, when I plug in the why that's gonna be 16 sine squared of Fada. Okay, Now, to find the to find Thea, what's going on to the boundaries? I need to find the derivative. This is only one variable, which makes finding the derivative fairly straightforward. So let's do it. That gives me 64 co sign of Fada Times Negative sign of Fada plus 32 times. Sign of Fada Times Co sign of Fada. Now let me just come up here. So I've got a little more room. I can simplify that. That gives me negative 32 Sign of Fada Times Co sign of Fada. So where does this equation equals? Zero. Because those are gonna be my critical points that I care about along the boundary. Well, either sign of faith equals zero or co sign of, say, the equal zero for Sign of Fada equals zero. I've got two options. Either say to equal zero or say that equals Hi. Now let's figure out our exes and wise when Sadie equals zero. If I plug that back into my values of X and Y over here, that gives me X equals four. And why equals zero if they'd is pie next zero, Why is for now? What if co sign of faith equals zero? Well, that also gives me two possibilities for Seita. Either. Theta is pi over two. Ortho is three pi over two and in each one of those theaters also corresponds with a point. If I plug those in for X and Y, I get X equals negative four and why is zero or X zero and why is negative four. So I have five points toe Look at my absolute minimum, and absolute maximum will occur somewhere at those points. So let's make a table. I'm gonna have five points. X equals negative for 00 negative four. And I want to compare that to the value of my function at that point. So what are those values. Will it? 00? The function equals zero. At 40 it's 32. At 04 it is 16 negative for zero, the value is 32 and it zero negative, for the value is 16. As you can see, I have an absolute maximum value that occurs it. Zero. And I'm sorry I wrote that wrong. It's an absolute minimum, obviously. And 32 is my absolute maximum. And it occurs to points along the boundary. So these are my critical points and the valley.

The problem is finding absolute maximum on many more layers of ash on the side. First, the computer. Partial half tracks I shall laugh for. Why in five is a critical point. The function off on the South D I should have cataracts. Is he going to two X minus two. Otherwise secret, full line minus sport. We like each of them. You go to zero. We have actually going to why you would want This is a critical point on the site on one one. They want to negative two. Then, well, Karen Value's off. Off on the boundary off the axe is equal to zero. Why is between zero and three War? Because, too true, why's between zero and three is equal to two y squared minus a Why plus one. This is equal to two times why modest one squire minus one Many more worry off, off. There's a connection. One nice in awhile. You off. It's the culture one. Why it before, too three has some Mike's more value. Seven. Why it's equal to zero x between zero and true. After you called your ex Goya minus two X, This is a cultural experience. One squire once one. So what actually is a good one? I've has a minimal audio. Next to one axe is you got to zero. It has a maximum awhile, Cyril. A lie is equal to three. Access between zero and two is equal to X minus one squire six. So when Max is a good one has minimal You next one, My axe is equal to zero. F has a maximal while you sell it. Then they compared these values half on the site Of the minimal value off half is the coach connective to the maximum value off off is equal to seven.

For this problem. We need to find some extreme values for the functioning over the region. So we need to discuss two different cases. Case one is the period eso we need to find a critical point. F sub X equals two for its and its sub y equals 2 to 1. If we set both pressure, curative equals to zero. We have X equals Y equals zero, which is inside this region. Eso At this point, f equals zero on for the boundary case, we use the ground multiply method. You find the function due to be X squared plus y squared minus 16 on we use the ground multiplier. It gives us a system of equation. Mhm. Okay, so we have several different cases from to solve this, uh, system of equations. So, for the way, can look at the second equation. So, either love. Nice. No. Zero. That means, uh, if Lambda is no zero, that means Lambda has to be watched safe. Loved I equals two one. We have four X equals two to eggs. That tells us X has to be zero. Then go back to this constraint equation. We have y equals two class minus four. That means we have two different points. P one equals 204 P two equals to zero minus four. And for both point f equals to 16. Not another case. Good. We can just sit like that across zero if lambda equals a zero. Uh, if Lambda equals zero, we have X equals. Why? Because zero, which is not the case because they're eo violates this constraint equation. Now, the third cases, we can actually, uh, we can actually said why, Toby zero safe. Why? To be zero, uh, with a constraint equation. Mhm. We have X equals two plus minus four. So it corresponds to different point. P three equals to 40 p four equals two minus 40 And the both point corresponds to f equals 32 0. Conclusion is, uh, this is the maximum value, and then this is the meaning. Um, better. Mhm. Mhm.


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