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# Find the critical numbers of the function.$g(\theta) = 4\theta - \tan\theta$

## All numbers of the form$\theta=2 n \pi \pm \frac{\pi}{3}$OR$\theta=2 n \pi \pm \frac{2 \pi}{3}$, where $n$ is any integer number, are critical numbers

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let's find the critical numbers of the functions she of theater equals four. Theater minus tangent to theater. Remember that Critical numbers or CC critical number of a function or let's say better this way. Uh huh. See in the domain of the function F is a critical number of F. If derivative of F. At that number does not exist Or it exists and is equal to zero. So this is our definition of critical number of a function. That is numbers in the domain of the function where the derivative does not exist or is equal serious. So the first thing we're gonna know about the functions to domain and now we consider the domain of G is equal to the domain of tangent of theater. Because the only restriction in the formula of G is given by the restrictions on tangent to theater. We know that is the real numbers minus this sort of hear the real numbers minus all the multiples of to buy plus by health and negative by us. That is two and by multiple of Dubai more or less by half. We're in the uh intelligent numbers. That is what that is. Because um we know that the tangent of an angle is not defined for the angle equal to buy half or negative by health. And then we have the period to buy. That is why we get this formula here. So we have all the real numbers except those of this form. And is there where we can find the so critical points of G. Then uh we got to find the derivative of key. We see that that's equal to four 2nd Square of Theatre. Oh And that's the same as 4 -1 over cool science square of theater. And we can see here that too derivative of She does not exist for theater Equal to where the coastline and zero. Well chosen. zero. If we look at the graph here of co sign we have here here Bye have coziness era at Syria's one and we have here five and 3 x half and so on, antique by house over here. And if I and so on. So we see that it's zero by half then three by half. So we can say that he is at the point by house plus Multiples of two by because the periodicity of the function. The sign of eggs is to buy. So yet we know that by half zero. Then we have all the other points from the to the right to the right and to the left and negative. I have also that is we can say More or less by health plus two M. Pie. But as you can see as we write this a little bit more Yeah within the multiple it by in front. So yet two emp I more or less by half. And as you can see these are just the points that are not in the domain of the function but these points these values of theater are not in the domain of jean. That is we cannot include these points of the form to and pi plus uh was more or less by half. We cannot include in the list of critical numbers because these points all these points are not Into the main energy and that's it. 1st. Uh requisite for The point of the critical point or # two. The critical numbers. So the only critical numbers of these functions are those volleys of theater for which the derivative in syria. So let's solve equation T derivative equal zero at least four minus one over cosine squared theta Equals zero. That is co sign square of theater equals 1/4. That is Co side of theatre is more or less 1/2 that's equation. So in fact we have two questions you have call center theater equal one half and co sanitary go negative one. How we look again at the coastline function But we are looking for here is Those fighters of the argument of cosign for which we had value 1/2 And for which we have to value negative 1/2 and uh we know the chosen is an even function. So if we know one value which is a solution without the negative value also is a solution this case. For example, for one half we know that this value here This solution we know because scientist 1/2 at the Angle Pi 3rd. So negative pressure by third will also be a solution. So The two Fundamental solutions for Go sign a part of theatre 1/2 are more or less by third. But then if we sum a multiple of two pi we get the other solutions. So this is that plus two and pie four inches and And for cosign a theoretical negative 1/2 we get the fundamental. Now we're looking here He had 1/2 here. And so the first solution is positive solutions is one here. It's easy to see that the solution is to buy. Uh huh and then that plus two empire give us all the solutions of this equation but we know that the negative 2.30 social solutions so we get all these formulas here. So all those values of theater having this form here or these forms here give us solutions to these equations. And are those the critical numbers of the functions E. So we can say that the critical numbers of she are see the equal mhm Or less by 3rd Plus two and by or They're equal more or less to buy three blessed to m pie or in any insurgent number. So this artie pretty cool numbers of the given function sheeting

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