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0 pts:) 6.) Three nubers add up to 6,5 [F two of these numbers are the same What I5 the largest product between the tWO nuhers that are different ?...

Question

0 pts:) 6.) Three nubers add up to 6,5 [F two of these numbers are the same What I5 the largest product between the tWO nuhers that are different ?

0 pts:) 6.) Three nubers add up to 6,5 [F two of these numbers are the same What I5 the largest product between the tWO nuhers that are different ?



Answers

Solve. One number exceeds another by five, and their product is 66 Find the numbers.

So in question 69 it says among old off the pairs of numbers, with some six Find the bear with the largest product. What is the problem? So let's assume that the two numbers are X and Y. So if I say for example, X plus Y equals six, we know that there's some six. We need the largest product. So if I just try to isolate why and make it Y equals negative X plus six, this is in your function, right? And now, since that we're talking about the product off the two numbers, which is X times Why? So I wanna multiply by exit to the whole thing, since I'm not talking only about why I'm talking about X y or the product that's equal to negative Excess Square plus six x, I just multiplied by X two all the terms here. So my function, which is the product I'm gonna name it f of x or the product off X and Y that's equal to negative X Square plus six X. This is my function, and the question is, what are the two numbers which make it the maximum? Ah, if we imagine that this is a problem. I need the verdict's year. The the why Valium, which makes this point the big point. So I'm gonna find the verdict's using this rule X equals negative B over to a and here in this case, negative view, which is negative six over double A into his negative one. So two times negative one. That gives me three. So the X value is three. We can just plug back here to find the Y value, so the Y value will be also three. So the point or the Vertex point will be three and three in dysfunction. Um, this is the bear in which it makes it six. And also it has the highest product. But he's asking here about the product itself. So the product is nine. You just multiply three Times Street or the two numbers. The highest product will be knowing

So we want to find what pair of numbers whose sum of six has the greatest product are some being A Plus B is equal to six. Our product is going to be used multiplication, which will be eight times B is going to require product. Mhm. So the first thing we want to do is we want to write out all the pairs that are going to be equal to six. So we will have zero. Press six one plus five. Do plus four And 3-plus 3. And the same for the reverse six plus 05 plus 14 plus two. Okay, reason being is when we use our multiplication, these were all end up equaling the same thing whether they're in reverse order or forward. So first thing we have zero times six. Anything Times zero is going to give us zero And we have one times five which is five. Next up we'll have two times four, Which is equal to eight. And finally three times 3 which is equal the nine. So if all of our sons that equal six. If we already use multiplication three times three, Equalling nine is our largest

Right. So you're getting another work of more told the difference between numbers next minus Why his father? We're also told that four times the larger number is six times the smaller number. We know the larger number of years X so or X equals 600 out numeral substitution. X equals five. That's why people again close y for X Gonna have 20 plus four equals six. Why, this is gonna give us 20 equals two wife, which means wife was 10 because we divide both sides by two. How we plug and White was 10 into this equation and we're gonna have X equals 15. We know what your numbers are assisting and

Okay, this is number six. We have two equations we have to satisfy. Um, if we are able to combine these, then we're gonna get a quadratic function, and we can figure out what the minimum is of it, right? Eso because we know this bottom. What is true? We also know that we can solve for why, right? So here we have the opposite of why is six minus X? Because we subtracted from both sides, Which would mean that why is equal to X minus six. Hope I didn't trick you there. I just that's you know, if you want to buy by negative one, that's what you get. Then we can take this guy, and we can substitute them for why we can get our quadratic. So we're gonna go ahead here and we're gonna have X times X minus six. Right s. So we know that Q, Which is are still I believe it's are, you know, just our maximum product here. It's our product in general. I guess we're looking for the minimum. Yeah, the minimum product on this one. So I'm going to say X squared minus six. Thanks. Okay, So if we think about this. I don't know where this is going to be, but this is gonna be some kind of parable it right And there's gonna be a spot right about here. That's going to be the minimum value for this. Whatever the X and y are of that, that's going to give us what we were looking for. How do we find that? Well, we know that the slope right is going to be like, you know, the slope is the derivative that's going to be equal to zero. So, you know, if we can find a spot where Q Prime is equals zero, then we have found that spot. We have found those points. So let's go ahead and say, What is Q Prime? We need to know what the derivative IHS. Luckily, it's a fairly easy one s o minus six on this one. So we set a sequel to zero, and we can go in and solve for X looks to I got carried away. Three is equal to X. Okay, that's half of the answers. So they were asking for what are the two? What we need to find. Why the easiest way to substitute this in is just because you know that X minus y it's six. We can just say that three. We know this is true with three minus y has got it also equal six. Right, Because threes are acts that we found s so we can go ahead and just say that, Um, So why is six minus three and we can say that why is equal to six months? Three is three. So this would be negative. Three. So our two numbers, which are an unaltered pair, doesn't matter what order these come in. Okay, You're still gonna work, so I want to write it like an unaltered pair. This is not like an ordered pair. Um, three on negative three. Can those our answers for number six?


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