Let $ f(x) = \tan x $. Show that $ f(0) = f(\pi) $ but there is no number $ c $ in $ (0, \pi) $ such that $ f'(c) = 0 $. Why does this not contradict Rolle's Theorem?
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all right. The question here is let every max equal tangent of actually that after dearly but Africa pie. But there's no number. See, Indira prices that have privacy You know what it is? Not contradict a belief system. Um Okay, so let's begin. I was first begin by sharing that effort Zero they could've effort pie So detention of zero So need putting f zero We get it Ah Qianjin of you and so the tangent of zero is if you recall remember the tangent function is just a sign over the co sign the loan you plug in sign of zero at the top This is really is a sign of zero coastline is you Coastline of zeros one but the sign of zero zero So this gives us a value of Syria And if you plug in effort pie You did the same thing Detention a pie And then we know that the sign of pie is also zero No And the coast on a pie I believe is negative one which still gives us a zero. So we know that after zero is equal in Africa Pie Who satisfies that conditions on DSO before we actually addressed continuity and different ability. It is really important to understand the graph of F. Brax s O if you recall detention of eggs is actually a is ah, it's not continuous function and the reason why it has to do with his pure just city. And so we know that the tension from sinners pie periodic. So it I had had the pi periodic and so there's a bit of a problem here. So the function sign over co sign X produces a problem because it is a rational function. If you think about it that way, it's a sign of a coastline. So when the coastline function is equal to zero, we get a zero at the bottom, which gives us A which is an undefined function and undefined function. Gives us a infinite a problem because his point where did not equal Tio So what this produces is a graph that looks actually like fists changing our backs. Remember, if you recall as she looked something like that and um, this occurs at all Interval. So this is negative Tyler too. And this is positive. Fly over to and this occurred every pie interval. So this is gonna keep going on. And this is also going to keep going on this way and you can see that this function is not continues. We have a vertical acid, so the continuity is does not exist here. Now the question of different ability is still here. So since this is ah function dead of pie periodic, it seems initially that it wouldn't be defensible but their pride. But if you look at the graph, Miss Asi from negative pirate too from pirate too. If we're going from zero to pi, we have to cross over this vertical acid toe. But we can't take a derivative at Pi over too. So this's also not a true statement. So since these two conditions fail, we know that this function tension of X does not satisfy roast the conditions required for roast them so does not contradict it.