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Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $ c $ that satisfy the conclusion of Rolle's Theorem.

$ f(x) = x + 1/x $, $ [\frac{1}{2}, 2] $

$$

c=1

$$

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Missouri State University

Campbell University

University of Michigan - Ann Arbor

all right. So verified that the function terrified the three parties is a broad term on the given interval. And then we're being asked to find all the numbers, sees that satisfying the conclusion of growth. So our first question is, whether is dysfunction continues because these are the three conditions required for roast room. And, um, if we look closely, we have a one over X and, um, one of her dysfunctions, actually not continuous at X equals zero. Could the few program X equals zero? You get zero plus one over zero. So you get Teo plus one over radio, and any number divided by zero does not. It goes off into positive or negative infinity, depending on where you coming in from. But however, since we are not in since our actual interval does not include zero, we don't even have to worry about this. So this all those true did not matter. So we know that it is continuous everywhere except X equals zero. So in this case is one half to two, which means it is also continuous and one half to two and then centralised, continuous. And from one half to two, we can also assume that is also defensible in this interval and then our last conditions, we have to track its whether Africa one half tickle toe too. And if you plug in the value of one, have you get two point five and if you plug it in for athletic too, you want to get two point five? So now that all three conditions of our function has been satisfied, we can now find the we can actually not find the value of number Seas have certified his conclusion. So all we have to do now is take the derivative of primary Rex terms at that equal to Teo. And since this is a rational function which is also polynomial, you can turn this into thanks. Plus, thanks to the minus one, since that is a property of a X exponents. And so this will give us one minus X minus two and we're being asked to set this equals zero. Will you bring accident negative two on boats to the right side. Since that is a negative and our experiment, we put it to the bottom and then we X squared equals one. And now we just take the scribble it on both side a squared of one. Get just two incher and get us positive and negative one, since our interval is all is in the positive side. So from one half to two we don't have to worry about negative ones. So R. C that satisfies this. It's just positive one.