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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^3 -2x^2 - 4x + 2 $, $ [-2, 2] $

$$

c=-\frac{2}{3}

$$

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Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Okay, So the question we're being asked to verify that the function of satisfying three high positive roles and begin interval. And then we're asked to find the number of sees the status by the conclusions of the rules doom. So the function here is after practical ex Cubans to X squared minus four plus two. So here we know that a Mexican ex humanity experiment for experts to is a polynomial. So we know that all polynomial is continuous and also are all defensible because those air properties upon our muse. So we know automatically now these first two conditions of roast and the satisfied and this is the last condition is where the end points are equal to each other. And if you actually plug in the values of of negative too, you get negative six. And if you plug it in for half of to, you also get negative six and you can check those values yourself and that again. Now that they're all satisfied, we can assume that the conclusion is also food in the conclusion there is a number. See, such that that there's a number c in the derivative of the function, so that it is. It is equal to zero. So we can actually solve for this function because we confined the derivative of affect. This is a very easy wanted to is very easy to do because it is a problem for me. So f Crime of X is equal to three x squared minus for X miners for and all we're doing now is asked to find where dysfunction equals zero. And this is a polynomial. So, uh, you can yes, you have to factor this. And so there are numerous ways to factory was function. I do it with the expended. You may have your own methods. So all you do is multiply ity and numbers are four time negative. Three times negative forgives us. Ah, negative twelve. And then we take the middle number, which is negative for. And then we find two numbers that multiply to give us negative phone. But add to give his naked for and in this case, in his negative six and positive too. And since this is a has a leading coefficient in front of the ah of ah, first number and and this polynomial we divide that way we divide that by the leading number. And then this gives us a simplified fraction of negative dory over one. That means we take the bottom number, which is one so be one X minus three times and this function is the most simplified fraction. So you just take the bottom for number in four acts in front of three X plus two vehicle two zero. This is the factored form of there are derivative. And now all we do is fine where these individuals terms equal dear, and they equal Geo when X is equal to three. I could go in the stall for this function in history, and then he saw for three exposed to it because three which gives us well, it gives us, I believe, negative two thirds police. So those are too, uh see or X values that give us where the derivatives equal to zero. And no, this is a very, very important point about rolls term. We're all storms only tells us that there is at least one see, so we know that there's a least one see for sure. But in this case, we have more than once that we have to, in fact, and this is an important point because all roads there tells us that there is one that there's at least one there could be more. Well, that could be one, two, three, four and infinitely many. And that is just an important point that should be understood of that role. Stone is that it proved that there's on ly that there's at least one if these conditions are satisfied.