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