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Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers $ c $ that satisfy the conclusion of the Mean value Theorem.

$ f(x) = 1/x $, $ [1, 3] $

$$

c=\sqrt{3}

$$

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Okay, so now we're being asked to verify that the function stratified to me about their monthly given interval. And then we're are being told to find all number sees that satisfy the conclusion of the mean value. So in order to know whether this function is continuous on wonder three, I think the best way to approach just one is to craft affects equals one of Rex. And so the graph of one over X looks something like this. It certainly does not defined it. Zero in Ashley goes off too positive infinity from the right and then to negative infinity from the left like that looks something like that and centre and of those only from like from one two, three was just probably approximately along this line right here. We know that just by looking at it, we know it's continuous, so it is definitely continue. And you can see that is also defensible because we can take attention at any point along this line, a wrong d's along this interval. So this is also differential. So now that we have satisfies the conditions for the mean value, then we can apply the virus. You can apply and find the conclusions of the medieval Gotham. And so the conclusion states that there is a number see such that and the derivative of every Becks that Point C will give us the same slope as the average float between between in the interval from wonder to be, which just means that effort to me. Minus half of one over three minus one. That is how you find the ever slope from one the one and with those apply this right away. So the value of efforts to be so we simply plug in effort three into original function in that gives us go on third and you can plug in one and that gives us one. And we can now apply this to our average slope, which will then give us. Ah, wondered. Minus one over two. And a derivative of one over X can be rewritten. So this remember that this function one of Rex can be re written as accident negative one, and you can take your do it if I bring this down. And then it's a plaque on this by one which will then give us. We're going to write this in terms of C So this gives us a negative one oversea squared. And now we're simply going to subtract one. So get negative one of the C squared equal to negative two thirds all over, too. I'm gonna bring this to the next page, and that is our negative two thirds over to just simply give us one third. So that is equal to negative one third. And now we just flip this and the negatives canceled. This just gives a C squared seacoast three and we take the square root of both side. And remember, squared gives us a plaster mining. This is plus or minus three three. And our interval, as you recall, is only from one to three sets all positive numbers. So the negative Route three actually is not included in our answer. So this is only part of the three. That is our answer.