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Kttaung " [ncn-Asing " local min fhas - kcel , at fbes . concuie conczvc doxn infkxtion E Pointfs) (has203 Ghn 40 on mitt tnc'5m fte) enan= 'oon4D "muei = FmJ L4 Ennntele ~hamf' Find Jct

kttaung " [ncn-Asing " local min fhas - kcel , at fbes . concuie conczvc doxn infkxtion E Pointfs) (has 203 Ghn 40 on mitt tnc'5m fte) enan= 'oon4D "muei = FmJ L4 Ennntele ~hamf' Find Jct



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Local max/min of $x^{x}$ Use analytical methods to find all local extreme points of the function $f(x)=x^{x},$ for $x>0 .$ Verify your work using a graphing utility.

So since we want to find out local extreme A. We will plug this in hair I'm in. See that? At 00 there's a graph, but we're more focused on what goes on over here. So at 2.718 we see that this is a local, um, maximum for the graph. Then for the next part, we want Teoh verify this and we see that right here. This would be the same if we zoom in here. This would just be a local minimum for the derivative. So the main point that we want to focus on is this maximum with the y value 1.445

So this problem. We're trying to help a political party which is planning a half hour television show. We want to maximize the number of viewers who are gonna be watching. Um and we're going to do that by changing the amount of time that three different people will be on screen. Ah, Governor, a congresswoman and a senator. So what is the equation that we're going to try to optimize? But we're told that for each minute the senator is on the air, they think they'll get 35,000 viewers. So 35 except one again, these numbers are gonna be in thousands of viewers. And just so we remember for later, X up one is going to be my senator. So every minute the senators on the air, uh, they think they're gonna get 35,000 viewers. Well, what about the congresswoman? Well, for every minute the congresswoman's on the air, they estimate they're going to get 40,000 viewers, so we'll call, except do. That's the number of minutes The congresswoman is going to be on air and third will finish with the governor. They think they'll be 45,000 viewers for every minute the governor is on the air. Okay, so that's the equation that we're going to try to optimize now. What are our constraints? Well, we know that the senator wants to be on screen at least twice asl ong as the governor. So the senator wants his time to be at least twice asl Ong as the governor. And I'm going to rewrite this because I would like to have this as a less than or equal to equation. So I could say this is negative X one plus two x up three is less than or equal to zero. Okay, what's my other constraint? Well, I know that all together, if I add up except one x up to and accept three, all of them together at most could be 27 minutes. It's a 30 minute program and at least three minutes are already gonna be dedicated to direct requests for money from viewers. So 27 minutes at most is left for these three people together. And our last constraint says that the total time taken by the senator and the governor so senator is except one plus the governor except three must be at least twice the time taken by the congresswoman. So at least two times, except to and again, I'm going to change this. So it's a less than or equal to format, so that becomes negative. X up one plus to accept to minus except three is less than or equal to zero. Okay, let me write thes three constraints over again, adding in a slush variable so I could turn them into an equality instead of an inequality. So the first one, except one plus x up to plus except three plus my first slush variable s sub one well equal 27. My second equation. We'll have my second slush variable. And so that will equal zero. And my third equation. We'll have my third slush variable s sub three and that one will also equal zero. Okay, so I have my three constraining equations. I have my equations. I want to optimize. I can set up my grid ready to do my simplex method to solve this. So I have three variables x 12 and three. I have three slush variables as sub one s up to SF three and I have ze so first constraint except one plus except two plus sf three plus s sub one. The rest of zeros equals 27. Second one negative X up one plus two except three plus s up to equal zero. Everything else gets a zero in there and my third one negative except one plus two, except to minus X up three plus s sub three equal zero and finally, on the bottom I'll put my Z function, and I'm gonna put everything to the left hand side so that I can set it equal to zero. And that gives me negative 35 x up one minus 40 except to minus 45 except three plus C equals zero. Okay, now we're ready to solve. So we went to look in our objective row at the bottom, find the most negative number, the largest negative. And I'm going to check the numbers in that row. Um, that sorry in that column except three column and compared to my Constance and see which ratio is the smallest Well, 27 divided by one is 27 zero. Divided by two is zero. And I'm not going to be looking at my negatives. So the smallest zero, which gives me. That's the second row there. So let's come down a little bit and we're going Thio, refresh our grid and do our next iteration. So the row with our pivot point to doesn't change. So I'm just gonna re copy that middle row, row number two. Now I want to make my goal is to make everything else in that column equal to zero. So let's look at the first row in order to make that one in the first row, equal to zero. I'm gonna put this over here and read so you can see it. I'm going to take the opposite of the second row, plus twice the first row, and there's going to be my new first row values all the way across my grid. So when I do that, my new row one looks like this 3 to 0 to negative 100 54 right now onto the third row. In order to get that third row equal to zero, I'm going to take the second row plus twice the third row, and those will be my new third row values. And when I do that, there's my new third row negative 340 01 200 Okay. And the last one I have to make that minus 45. I have to change that to zero as well. In order to do that, I will end up with 45 times the second row, plus twice the fourth row. And those air. My new fourth row values in doing that gives me a new fourth row of negative 115 negative. 80 00 45 0 to 0. Okay, I still have negatives in that bottom row, so I'm not done. Eyes look and again find the largest negative number. And I'm going to examine every value in that column and compare it to my Constance to see which one is the smallest ratio. Well, this one's pretty easy. I don't wanna look at my negative numbers, so I only have one possibility, and that's that three. That three in the first row is going to be my new pivot point. So But right there re copy my grid ready for my next iteration because my pivot point is in the first road. That's not going to change. So I can re copy that first road just the way it is. Okay, Now we want to get rid of every other value in the same column with my pivot point. So row to I want to get rid of that negative one. So I will take three times the second row and add that to the first row. And that's going to be my new second row number. When I do that, my second row is gonna look like this. 0 to 6, 2200 54. Okay, on to the third row. I want to take that negative three and make it a zero. In order to do that, I'll take the first row plus the third row. Nice simple calculations this time. That gives me a new third row of 06 02 0 to 0 and again, 54. Okay, last one. I want to get that, um, very bottom row. I want that negative 1 15 to now be zero. And to do that is going to be 115 times the first row, plus three times the fourth row. And those air my new four throw numbers. So my new fourth row is zero negative. 10 zero 230 2006 and 6210. Well, we are not done. We still have one more negative. So do exactly the same thing. Compare the values in that column to our Constance. First row 54 divided by two is 27 second row, 54 divided by two is 27 and then 54. Divided by six is 99 is my smallest ratio. So that six is my new pivot point. Hey, one more time with my grid. Hey, third row is my has my pivot point, so it does not change. I can copy it all the way across. Okay, To get rid of the two in the first row, I want Member. I want all of these other values in that column to be zero. So to make that 20 I'm going to take the opposite of the third row, plus three times the first row, those air, my new first row values. That gives me a first row of 9004 Negative three Negative too. Zero and 108 Okay, second row again. I'm getting rid of a two. So it's gonna be the exact same formula except instead of looking at row one I'm now looking at road to. So that gives me a row. 200 18 46 Negative, too zero and 108 and my last row. I need to get that negative 10 to become a zero. In order to do that, I'll take five times the third row, plus three times the fourth row. And that will be my new four throw values. And that gives me a row of 000 700 60 10 18 and 18,900. Hey, we are now done. There are no negatives on the bottom row, so I'm just gonna look for every column that has one and Onley one non zero value. Okay, so what does this give me? This says, except one. Well, nine x sub one equals 108 or accept one equals 12 except to Well, if I look at that column x six, except to is 54. So, except to is nine and for except 3 18 times except three is 108 So except three is six and for Z 18 times E is 18,900 So Z is 1050. So what do those numbers mean? Well, if we go all the way back to the top, remember, the except one was our senator, except to is the congresswoman and accept three is our governor. So coming back down to look at our results. Just we want to give the senator 12 minutes of air time. We want to give the congresswoman nine minutes of air time and the governor is going to get six minutes of air. Time and Z, which is our viewing audience in thousands, is 1050. So multiplying it by 1000 means I will have maximizing my viewers at 1,050,000.

Were given the graph Y is equal to the square root of X. And we wish to find the point on that graph that is closest to the 0.0.4 comma zero. So again we're going to use the same trick instead of minimizing the distance, We're going to minimize the square of the distance. So we don't have to deal with the square root. Okay, So we get D. Of X is equal to X -4 Squared plus. Why minus zero squared? But why is just square vertex? So it's square root of X squared. Okay, so that becomes we'll leave this as x minus four squared plus the square root X squared. Well, that is just absolute value of X. Okay, so now we can find the derivative. So the prime of X. That gives us two times x -4 plus during the vex. That's just what? Thanks. So now We want to set this equal to zero and the sulfur X. So we get zero is equal two. This is two X minus eight plus one, Which is just two. X -7. Okay, so then solving for X will give us X is equal to 7/2. Okay, so now we just need to again double check and make sure that this is in fact a minimum. So we take the double primex which is just to And this is greater than zero. So which means it's concave up and concave up means are critical value is in fact a negative. Sorry of minimum. So therefore our final answer is going to be um minimum distance from Y equals x two for like zero is the point X value 7/2, which means why value is the square root of seven or?

For this problem. We want to identify the critical point and then determine the local extreme values. So we're given here. Why equals X two The two thirds Times X plus Tim, Um, and one thing that we could do first is to distribute this. This will make things much easier in the long run. So we actually can, right? This is why equals X to the I'll be five thirds. Since we multiply, we add the exponents plus two x two, the two thirds. Now we can differentiate this and we'll get five thirds X to the two thirds plus, um, four thirds because we have two thirds times too x to the negative one third. And then what we'll dio is we will set this equal to zero. So one helpful way that we can figure out what this is is weaken. Graph it so we can start off with our original graph two thirds, uh, Times X plus two. Or you can go straight into the graph that we already have, which we just soft for. So when you do this, we see When it comes to the critical points, there is one critical point, and it occurs at a negative 0.8 So when we solve for this, we end up getting that X equals a negative 0.8. So with that in mind, that would be our critical point. The next thing we're asked to do is determine local extreme values. So what we see is that there's also a point where it's undefined. So at X equals zero, we know it's undefined because we would get zero in the denominator as a result of that negative exponents. So because this gives us, um, undefined or does not exist, we know that this is not a local extreme value, but it is a critical point. However, this right here is a local extreme value. If we look at the values before whether on the graph or, um plugging in values, if we say look at negative one, what we end up getting as a result is a lesser value. And then if we look at something like zero or actually will do something like negative 05 what we end up getting is, ah, lower value as well. So what we know as a result of that is that this is a local maximum


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