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Prove that the expression $x-2-x^{2}$ is negative for all real values of $x$...

Question

Prove that the expression $x-2-x^{2}$ is negative for all real values of $x$

Prove that the expression $x-2-x^{2}$ is negative for all real values of $x$



Answers

If
$ f(x) = \left\{





\begin{array}{ll}







x^2
& \mbox{if
$ x $ is rational}\\







0


& \mbox{if
$ x $ is irrational}





\end{array} \right.$
prove that $ \displaystyle \lim_{x \to 0}f(x) = 0 $.

This question asks us to prove that the derivative of co tangent acts is negative. Cassie can't squared X. We're gonna be using the quotient rule for this, which is F one G minus F G one over G squared. In other words, we're gonna be differentiating co Tension X, which is equivalent to coastline of acts over sign of X. So we're gonna be differentiating co sign over sign. Okay, First off sign stays the same. The derivative of coastline is negative. Sign of acts were then subtracting co sign of X. The derivative of sign is co sign of Axe divided by G squared. This is sine squared of axe. This simplifies to be negative one over sine squared of acts which is the same thing as negative Kaseke unscored of acts. The reason wise because one over sign is Cassie can't therefore negative one over Science Court of X is negative. Costigan scored of X. Therefore, we have now proven both side

All right. So the question that we're gonna be looking at today is we need to prove that the derivative this D over DX is live mons notation for derivative D over dx co tangent acts is indeed equal to negative. Go. Sorry, coasts. He can't Oh, cosi can't squared X. So we need to prove that the dirt of cocaine gen X is indeed equal to the negative. Cosi can't Squared X. So our first step that we want to do is we want to make, um we want to use the reciprocal identity code. Change in X is equal to one over Tan X. That's what we're going to do after we do this. That's just something that you kind of just have to know. That coat change in X is the inverse of Tan X. So now that we have that we can use the derivative of one over Tan X now that we have that we can actually put this into the product rule. Using one is f and tan as she So once we do this, we do 10 x time zero, which is the derivative of one plus one times the derivative tan X which is C can't squared X all over tan X squared. So now that we have this product, roll it out, we can actually simplify everything using some trig identities and some just collecting, like terms and such. Once we do that, we get negative. C can't squared acts over tan squared X can squared X So now that we have this, um, sorry. So make that. Yeah. So now we have this negative. C can't squared X over 10 x. What can we do? Well, we can actually split it up into fractions because what this is telling us, really is that, um it is really, um negative. C can't squared X over one our Sorry, not over times 1/10 x can square decks. So very good squared X squared. So now we can use the reciprocal theory be used before and simply just write it as that's negative. Seeking squared x Times co tangent co tangents Squared X. And now what we can also do with this is we're going to switch this up again and make c can't squared, um, equal what it really is, which is one over coast squared X and then multiply that by, um, Khost squared X over sign X, which is another way to write co tangent squared X And that's just on identity that you kind of have to know. And when we combine these two things together, we're actually going to get, um, negative. Cosi can't squared X. So in the end, we did get what we wanted at the beginning, which was CO c Can't squared X. So now I'm discovering a little, therefore statement, and we're just going to write that in conclusion, the Serie votive of, um derivative of co Tangent X is indeed equal to negative. Cosi can't x squared, and we're gonna right a little box or q e d to signify that the proof is done.

We need to prove that the X plus one over banks were greater in Could you chew? And in it you prove this one. Now, let's consider the angst. Ah, half minus expo minus 1/2 square. And we know this grand. Ah, scrambled difference. We're always great to ico ju zero. Ah, let me right on the on the side here and now by former phoned expansion I'm doing about you, which you get an extra half square minus two attempts Expert half times when the expert minus half. And now plus the expo minus 1/2 square Greater ico Jizo. And here for this one which get echoed your banks miners to this Jew handling it could you one. And now plus four doesn't get any coaching. The ah Explorer after minus one. Got a coach's Oh, he had about off minus one will be written down is the X plus one of ah, angst. And from this one here, I will bring it to the right inside. Now, if I should get express one of our ex worker to equal Jew Joe and examine anyone we need to prove here

Okay, Okay. We won't show. This function has really zero between the values one and two on we have to do to show. That s so there's a change in sign between these two numbers when input them into our function. Okay, so let's show this thing. Wanted to hit one key to 11 minus two, minus one minus two. Tight, isn't it? Two kids ate eight minus two times two. Well, eight minus four is full months. One to give you three on here. There's a change of signs. Sure, that somewhere between here is a real zero. Thank you.


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