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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.$f(x) = 2x^2 - 3x + 1$, $[0, 2]$

## $f(x)=2 x^{2}-3 x+1,[0,2] . \quad f$ is continuous on [0,2] and differentiable on (0,2) since polynomials are continuous anddifferentiable on $\mathbb{R}, \quad f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \Leftrightarrow 4 c-3=\frac{f(2)-f(0)}{2-0}=\frac{3-1}{2}=1 \Leftrightarrow 4 c=4 \Leftrightarrow c=1,$ whichis in (0,2)

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Okay, so now we're being asked to verify that the function f satisfies two hypothesis of the mean Barry doing on the Cuban interval. And then we're supposed to find all the numbers sees that satisfies the conclusion of the mean value they're here. We call that affects because to explore monastery experts one on the closing of war from zero to. So since dysfunction is a paranormal, we know automatically that it is continuous on the clothes. Wonderful was there too. And it was also defendable on Jura Institution That is a property of a polynomial function. So So now that this satisfies two two conditions required for Rose there, I mean for the mean value there. I apologize. We can test the conclusion. But I mean right there. And she'll just recall the conclusion for the main fire demons that there exists a number see, such that is the derivative of prime of C. There is a value where the average slope from A to B equals that there's a sea that give this a tangent line that had the same slope. So now that this satisfies this conditions, R B is too in our eight zero so the first thing I'm going to do is figure out what effort to is. And if you plug in your number into the function, do you get three? What? And half of one ISS. I mean, I'm I'm sorry. Half of zero is just equal to warn. So now I'm going to plug in all these numbers into my mean various room. The prime of C. We're going to come back to this ever be is in this case, three minus F zero, which is one what over two minus zero was then gives it to over too, which is simply one. And now all we have to do is now take the derivative of this function in terms of seeing. So now it's going to be Ah, so just do it like you regularly do. So did you ever have of axis for sea? Minus three is equal to one. So now we're looking for a sea that satisfies this condition. And now well, move over here. We're going to add the three to this side. Three, two, both side, and then we're going to get foresee is equal to for was Jen, give this see of one that is the number at us. They see a satisfying conclusion of the mean guy item, and in this case there's only one seat. So that is the correct answer.

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