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$23-36=$ Find the critical numbers of the function.$$g(y)= rac{y-1}{y^{2}-y+1}$$...

Question

$23-36=$ Find the critical numbers of the function.$$g(y)= rac{y-1}{y^{2}-y+1}$$

$23-36=$ Find the critical numbers of the function. $$g(y)=\frac{y-1}{y^{2}-y+1}$$



Answers

$23-36=$ Find the critical numbers of the function.
$$g(y)=\frac{y-1}{y^{2}-y+1}$$

Okay, let's take the derivative using the question rule, which is GF Prime minus half g prime over g squared. In other words, over the denominator squared. We do that. We end up with negative y squared. Plus two. Why? For y squared minus y plus one squared. Simplify this. We end up with negative y squared plus two. Why, doctor? Thus negative y terms. Why mine is too. We end up with two solutions. Weikel zero y equals two.

It's Clara. So when you read here, so we're gonna find the critical values. And to do that, we're going to get the derivative so derivative is equal to X plus two over three times X to the 5/3 we have to set of equal to zero. So this is when the first year of it of equals to zero. So the derivative does not exist when the denominator is zero. The denominator is zero when X equals zero and the new Raider is zero. X equals negative, too. So are critical. Values are zero and negative, too.

Problem 33 asks us to find the critical numbers of G f t equals T to the fourth plus t cubed plus T squared post one. Remember, the critical numbers are values of T that such that g prime of T equals zero or does not exist. So the first thing we want to do is notice that this is 1/4 degree polynomial and we know that polynomial czar differential everywhere. So in this case, there won't be any values of T where the which he promised he doesn't exist. So our goal is to find just a nice, uh, t where G Parma t equals era. Um, and the other thing to note about having 1/4 degree polynomial is that it's defined for all real numbers. The domain is all rials, so that can help us validate our answers. So the next step is to simply evaluate G Prime A T. And so we have a some of powers of tea. So we're going to apply the power rule teach term. 1st 1 we bring down the four and 40. Cute. Just subtracting one from the exponent T queued gives us three t square. He's squared is this to t? That's the power of one and the constant one. It's derivative is zero. So we'll just admit that term. And now we have a cubic polynomial and t here that we want to set equal to zero is there equals 40 cube because three K squared plus two t and I wouldn't want toe when we want to find the values of t T. For which this parliament or equal to zero. We'll need to factor this expression and set each factor equal to zero. So right off the bat, we can see that each term has a factor of tea. So hold that are, and in a tee times for a T squared plus three t plus two. And now we set each factor equal to zero. So for the 1st 1 case one we just have t equals zero. And there's nothing left to do their eso when t equals zero that gives us two prime of T equals zero. That's our first critical number, and case two this wind. The other factor, 40 squared plus three t plus two is equal to zero, and now there isn't an obvious way to factor this just by looking at it. So we're going to apply that quarter it formula. Where for is a very mole. Three is B and two C. So that gives us t equals negative B, which native three plus or minus the square root of B squared, which is three squared, minus four a. C, which is four times for times him all over two A, which is two times four. And now let's just start by simplifying the radical here. So three squared gives us nine. Four times four is 16 times two is 32. And so nine months 32 is negative. 23. And we have a negative number under the radical, which means that this expression I only evaluates two complex numbers and because the domain of our function is real numbers and we're looking for values of T in the domain of the function, we can just reject this case altogether. There's no more solutions that will come from, um, evaluating this expression, uh, so we can just write her answer as the set that only contains the number zero and we're done. So zero is our only critical number

We're given the function g of why equals why minus one divided by y squared minus y plus one. And our task with this function is to find the critical numbers of that function. Well, that's very small. Well, imagine that this were who is larger area critical numbers. We want to find the critical members of this function and recall that define critical numbers. You need to take one derivative of the function so we want g prime of why and set it equal to zero or take g prime of y and find when it does not exist. And any value why that satisfies either of these conditions is going to be a critical number of that function. So we want to start with this function g of y and find its derivative g prime of y, then set that derivative equal to zero. And so, for why and also find figure out the values of why the because g prime not to exist. DNE does not exist. We're going to start with finding the derivative. So we want to get G prime of why we're taking a derivative with respect. Why? Because why is the independent variable and because this is a rational function of the form, uh, I write this on site we got G equals some numerator. A divided by a denominator. Be G prime is going to be found using the quotient rule the question rule For a over B, it's to remind you as a prime a times B minus a times the prime divided by B squared. So this is the pattern that we're going to use to figure out the derivative of the function. The derivative of the numerator is one multiplied by the denominator y squared minus y plus one and then subtract the numerator. Why, minus one times the derivative of the denominator. The derivative of the denominator is too. Why minus one, then divide by the denominator squared. That's why squared minus why plus one that whole quantity squared. We're going to simplify this result, but before we do, let's think about what we want to do with this function after we simplify it so we don't do anything unnecessary. We're going to want to figure out when this function is equal to zero, and when the function doesn't exist for a rational function, a fraction like this it's going to equal zero when the numerator is zero and you're going to get does not exist when the denominator is zero. The reason is if you have zero on top and a number on bottom zero divided by a number gives you zero is the answer. If you have a number on top and zero on bottom, a number over zero does not exist if you have zero on the top and the bottom, that also doesn't exist as a result. But the way in which it doesn't exist is a little more complicated. We're not gonna run in that right now, so we'll say that for a different day. Simplifying this. I don't really need to expand the denominator. It's already a single term. When I set that equal to zero, they just stay as it is. I just want toe combine like terms of the numerator. It's the numerator we get. Why squared minus y plus one minus. I'm gonna foil these three terms together. Yet two y squared, Uh, minus two Y minus one Y combines the minus three y and negative one times one, It is a positive one that this whole term is being subtracted. So I'm going to have to distribute the minus. Sign each of these terms as I simplify and the denominators is the same White Square minus y plus one squared. We will get, uh, y squared minus two y squared people's negative life squared Negative. Why minus negative. Three y That's negative. White plus three y positive to why positive one minus positive One equals zero and nominators still y squared minus by plus one squared secures our G prime derivatives. And first to get G prime of white Will zero. That will happen when the numerator is zero. We'll have our first case. G Prime equals zero. We're looking at negative y squared plus two y equals zero. This is going to happen at factor out. Why here yet? Why? Times negative. Why wants to equals zero And we have solutions at y equals zero And why equals positive too. So these air our 1st 2 answers at y calls their own Michael's too. Do you prime it? Zero. That gives us two critical numbers. Second case we want to see When are derivative. G prime does not exist. It won't exist if the denominator equal zero So you want. Take this denominator term. Why squared minus y plus one set equal to zero and that the solve this. This is not obviously factor, Bols. We're gonna use the quadratic formula, which is negative. B running out of room here. Negative D plus or minus the square root B squared minus four a c invited by to a and those There are solutions to this quadratic set equal to zero and you'll see pretty quickly. We played this in, uh, we got a B C or one negative one positive one one plus or minus. The square root of one minus four times one times one one minus for was going to be a negative number in the square roots one plus or minus square root of negative three. Bye bye to these are imaginary answers. So there's there's no solution. There are no real solutions, the denominator equaling zero. So for the green part here, there's no rial solution. Seoul for solution on The practical implication of this is just that this original function it's denominator cannot equal zero. It's a rational function, but it has no vertical assam totes. So the first derivative, not existing one indicator of that is when you get Infinite Slope, you have a function that goes up to infinity. That happens with a vertical ascent. Oh, but this function does not have vertical acid cuts, so there is no contribution to the solution From the second part. We just have box and red over here. Weichel zero y equals two And those your critical numbers.


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