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Find the extrema of the function $f(x, y)=x^{2}+y^{2}-4 x-6 y+2$ defined over the region bounded by $y=0, x=0$, and $x+y=5$, see Figure Ex. 43.

$$\operatorname{Max}=7 \min =-11$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

15:30

Find the absolute extrema …

we're going to find the absolute extreme of the function. F X y equals X squared minus three y square minus two x plus six y on the close region bounded but the square with Curtis's Syria 00222 and 20 these regions is catch. Here we get the four vs here. Zero. Here it is here 22 and Syria to we had the reason is the square, including the battery. And we know that on that region we gotta have a natural vaccine, Mom on an actual minimum for the fun shit, because it is continues and the region is bounded and closed so we can have those Strama happening adds grateful points inside the region or on the battery. So we get a fine first regal point of, and for that we calculate the first order partial derivatives. So this partial derivative of Air Perspective X is equal to two eggs minus two and diversity derivative of if respect to why is think of the six, why policies? And now we get to solve the following system to find the griddle. Points to explain is to equal cereal and negative six y plus six equals zero. And from here we get immediately the exit one, and why equals one? So the only critical point, uh, off this function is 11 and it happened to be inside the square. In fact, it's the center of the square, so it's appointed. You can see there and there. At that point, the image of F is equal to one square. My 31 square from minus two. What Two terms? One, uh, plus six terms. One. And that's he would want minus three minutes to policies. That is too. So if at the critical point which is inside, our is equal to value to. So we get this fella here and we're now going to study the function on the boundary. So we start with these side here, and this eye is define is like will cereal an eggs ranging or going through zero and to and they're on their the function of becomes a function one wearable X, because why becomes equal to see room and it resisted is why able to Syrian this is formula. Here we get X square minus tricks. So that's the expression of F on decide, which is part of a mentor that can we factor out as X Times X minus two. So we have a terrible up. Who's graph is smallest like this you have. See, we're here one here, two here and the Brolin's have been upwards because he has a positive sign. The coefficient of X square. So we have something like this. So we're going to drawing goal or six c. So Curtis is here and here and probably something like that. And we know that the a move value is a teen at one. So we have here to maximum value zero because it's value. Detain ity burgesses here on two. Otherwise, the proble attained the values and the minimum something I didn't say. But it's very important to remember is that x ranges from Syria to to. So is the only portion of the problem. We get a draw now the mean value there was. You know that happens at one. So we have one times one minus two, and that is negative. So these are two values we got to concede There. Now let's talk about these boundary here and this idea. These define us what he calls, too. Why people's to four points in the segment and explaining from serious to to. So we have the second part of the battery. Why was too next going from Syria to to? And then the function becomes a function of X Only we're going to replace. Why by the value to here get X squared minus three times. Why square? So you get two square minus two eggs plus six What that is, plus six times too. And that's equal to X square minus four time 3 12 minus two eggs plus 12 On these two terms, cancel out and this is equal to X square minus two X. So we have a second this same equation we obtained before for these part of TV Battery and X is running under scene, he intervals. So we have identical situation. So we know already that we have the maximum value zero on the minimum value of negative aura. So it's being simplified your bed. So you have now the third part of the Bedri. We're going to consider these under here and his father here to find his ex able to Ciro on Why run in from theory zero to to So we have. Why X equals seriously. I'm wife for him. C 0 to 2, and then the function becomes a function of why? Only because we're going to replace zero. Uh, please. Xo yet? Negative three. Why square plus six. Why? And that could be a factor out. Three wide times two minus. What? So again, we have a terrible, uh, open downwards because coefficient of y squares negative pan the burgesses thes zeros. Our city is zero an too. So we had more or less disc. Raph. So we have cereal and to and he's open upwards. It's It must be like this more or less. It's very hard to do very meet trouble here. Okay, I would say that. And we know that the maximum value of these horrible over its hero to which is a range of value that we get for the viral in this case. Why? And he's from Sirio, goals from Syria through the value too. So we know here that the maximum value up he's ah, peanut one that is three times one times two minutes. Wanted his three on the minimum value is Syria, which happens at 2.0 and two. So we have, uh, these other two values. And now the fourth and final part of the boundary is X equal to and why from cereal two and that bars around the wrists. What so exists cost a table to to and why runs from Ciro through the value, too. And then again, the function becomes function of white only, and the formula it's obtained by replacing X by two. So we get X squared and it's two square minus three y square. Then we have, um, they have two times X. That is negative. Two times to blast six. Why? And so we have four minus three way square minus or six. Why? So these 24 cancel out and becomes thing Achieve three y squared plus six y for Why running for a zero through to And this is the same expressions before we can see here stretching sickness Samos it horn and the variable. Why lives in the same interval us in this case. So we have exactly the same situation. So we know ready. Take the maximum value his three. But here and in many a moon fattier he's serum. So now we're ready to say what are the streamer off the function If off X y So the absolute meaning, Mom, if all right is every year look at the minimum we have obtained in all cases. So we have hero here. See you here. Here we get near the one like here, So near the one he's at least at this moment, the smaller small its value of to function. And we have at this crucial point developed use of the minimum peas. The absolute minimum hits nearly one and the absolute maximum if if if more he's so we get the maximum. So we have here three. Three again Ciro syrups or three his for the moment e m absolute maximum. And here we have two. So three he's in fact the absolute maximum of f one are. So if the extreme of the function over or on the square to find about these four verses

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