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Problem 76 Hard Difficulty

For what values of $ a $ and $ b $ is $ (2, 2.5) $ an inflection point of the curve $ x^2 y + ax + by = 0 $? What additional inflection points does the curve have?

Answer

$a=\frac{-20}{3}, b=\frac{4}{3}$

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Video Transcript

Oh, careful what values of N B is to comic two point five, infecting point of the crew and square y plus ankles B y. And so, uh, to, in order to find the function that satisfies the A and B that satisfies where inflection point of two coming to point five occurs, Um, we have to first. Well, we First of all, let's start up with isolating why in this case. So we have Explorer wife. Let's take those b y. So remove the move The ex over to the other side will be negative egg, and then I can pull out Why. So this will be why X square plus be people never react. Sorry about that. Then this will give us why is equal to negative, eh? Over x squared plus b and then now convicted two revolutions. We want to get to the second word because we know that an inflection point. That means that the second derivative equal to zero. So we know that I actually did really well so be it cannot be X square minus a minus D a over x square plus be this is going to be squared and then the second derivative. Why? Double crime is equal tio to a act Times three B minus X word over X Where Class B. It's going to be cute. So now we're going to do is way. No, we have these two value. So we're gonna plug him in two our original functioning itjust dysfunction to see how we can use that later on to find the values of A and B And you'LL see it become more clear. So this is going to be okay. Sorry about that. This is going to be two squared times two point five and then I'll be plus a time too plus B times two point five. And this is because of you because that's what the original function is. And this comes out to be for a Perth five b less twenty the record too. So so reminder that I said that in order for an inflection point occur at this point, we know that the second derivative testicle d'Oh and we're going to be looking at where the numerator equals zero up to x and Wei have values of x and Y and Z complex is in. So too, eh turned to and then this will be a time three B minus two squared and that this we're solving for where the sequel syrup. But that's where the inflection point is. And so what this comes out to be for a three, the minus for and so the values in which just gives a zero r equals zero. Could you plug in zero or we get zero and people people going for be plugged in for third. You also get here, So these aren't actually the point. These aren't the phone numbers for our function, but we can use these numbers A and B to figure out the real agent for the Andes that we're looking for. What on? That's why we're using Ah, I'm sorry. Dysfunction to plug it in. So now that we have AIDS and bees, we can actually go ahead and put those in. It's all for our each individual immunity. So for the first one, I'm gonna put it in red. We're going to use a claw zero. So that means it's going to be, uh, to make this more clear, meningitis out before times, you know, Class five B, which we are not substituting on, plus twenty This is only clues. You know, he saw for being could be coordinated for now. If I do it for B is equal to for third. What's gonna happen to me for a It's fine kind for third plus twenty crazy. And this will come out to be a good negative. Twenty third. And those are two. Sorry, this is not clear. Negative. Twenty third and B was not gonna afford These are actual paid and bees of our function, and that's it.