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Maximize B = 5xy? , where x and y are positive numbers such that x+y? 10. The maximum value of B is (Simplify your answer Type an exact answer using radicals needed...

Question

Maximize B = 5xy? , where x and y are positive numbers such that x+y? 10. The maximum value of B is (Simplify your answer Type an exact answer using radicals needed )

Maximize B = 5xy? , where x and y are positive numbers such that x+y? 10. The maximum value of B is (Simplify your answer Type an exact answer using radicals needed )



Answers

Find the positive values of $x, y,$ and $z$ that maximize $Q=x y z,$ if $x+y=1$ and $y+z=2 .$ What is this maximum value?

Realization problem. We're trying maximize Q. This restraint function Over here. So let's go ahead and isolate the easiest one. Isolate Florist for why? So what do Y. is equal to two -X. And then go ahead, proceed to play that in. So we have X squared. Attempt to minus X. And then distributing that. We have two X squared -X. to the 3rd power. It's equal to R. Q. Here. Going and derive that. So we get four x -3 x squared. Pulling out an X. Here, we got four minus three X. Set that equal to zero here and well we get our two possibilities here. of x equals zero And 4 -3 x equals zero. So we're gonna ignore this one here. This one would give us three x equals negative negative three X equals negative four. Therefore X equals for over three here. Okay, so that means that when we go ahead and plug this in, so we've got my equals two Over 1 -4/3, it's going to be equivalent to six thirds minus four thirds and she calls two thirds here. Okay, so that's your answer here, so we've got X. is equal to 4/3 uh huh. Is equal to 2/3.

Okay for number seven, we have two conditions we have to satisfy, and we're going to find the maximum on this. We're looking for a maximum. So we have a situation where the graph would look something like this, and we're looking for this point like, where is that point in terms of X And Y, um, we know that if we can find something that satisfies both of these, they were gonna be looking for a spot where the derivatives equal to zero. And we know that's the slope. Right? So we're looking for Q prime equaling zero. Okay, Since we have these two givens over here, I'm gonna put these in terms of one another, right? I'm going to say that, actually, let's go ahead and solve for a while on this one. We're gonna say that why? Squared is equal to one minus X. Okay, I just subtracted X from both sides. The reason I do this because it works nicely, can substitute back into the first equation and have something that's all in terms of X, something that we can deal with. You could do it with the opposite variables, but turns out it's kind of hard because you got a big square root in the middle of it. So we're going to do it this way. Um, substituting back in, I have X times one minus x so I could multiply that through. And I actually have X squared Negative X squared plus x. Okay, so that's gonna be both our Q Again, we're looking for where? The derivative of zero. So if you could take the derivative Q prime of all that mess, we end up with negative two x plus one and thats going thio b zero at the point that we're looking for because at the maximum zero. So if he said it to zero, we could go and solve for X, and that's going to give us X equals one half. So we know what the X value is right now. Remember, we've got a situation where, um, we can scrape something back in and find the answer for why? So let's go ahead and do exactly that. That's one of the numbers, and this could go again in any order. The order is not important, but let's go and use this equation that they gave us right here. Let's go ahead and say that one half plus why squared equals one. Subtract one half from both sides. Right. So we have minus a half minus a half, because this one half So we know that why squared equals one half. Well, that means that we have Let's go down a little bit. Actually, two answers, right? We have the square root of both sides. OK, so it's gonna be plus or minus the square root of one half. So why is equal to this? That actually gives us three numbers in our solution set. Um, you could write this another way, but I'm gonna go and just keep it like that, just for ease of writing. So our final solution ends up being he ordered said of one half and let's just call it one thing.

For this problem we are asked to use lagrange multipliers to maximize F of X. Y equals X. Y. Under the constraint that X plus Y equals 10. So what we want is to have the gradient of our function X. Y equal lambda times the gradient of X plus Y. So from that we'll get the system of equations that why must equal lambda and X must equal lambda And then x plus y must equal 10. So we can see pretty easily then that we get that to lambda equals 10. So lambda equals 1/5, not 1/5. Excuse me, Lambda equals five. So we get that X equals Y. And they both equal five. Which then means that the maximum value F. Of X. Or the maximum value is going to be F of 55 Which equals five times 5 equals 25, giving us our maximum

This video. We are going to fight in maximum value off this function f when restrict to this graph X plus Y minus two equals zero. So we have function G as this form. Then we look at the radiance equation. Great in F equals, uh, Constanta nhs. Grady, enough. G u May have to use a shan drew here for for the great enough, But it shouldn't be too hard. They are politically, after all and grading off. Jeez, as simple as this. Now, comparing the core vision off ish coordinate i n j. Well, give us this system, then. That is equals. Who? My last two eggs over. I gonna call this a just for short. We won't be using them so minus to why I overdid it. And this already tails out this at the point of contact, we have x equal. Why, right? Because they are off the same four. Now, we put this in our restriction function. So it plus y you close? Ooh. Then it means to wise equals two. So why is one and so the stakes? Our function is six miners x squared, minus y square and square root. So we put X and y equals who want we get? This guy rode a fault, which is two. So here we have it. This is the maximum if when restrict to this craft. Uh, let's go back here is too. Thank you.


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