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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) = 2x^2 -4x + 5 $, $ [-1, 3] $

$f(x)=2 x^{2}-4 x+5,[-1,3] . \quad f$ is a polynomial, so it's continuous and differentiable on $\mathbb{R},$ and hence, continuouson [-1,3] and differentiable on $(-1,3) .$ since $f(-1)=11$ and $f(3)=11, f$ satisfies all the hypotheses of Rolle'sTheorem. $f^{\prime}(c)=4 c-4$ and $f^{\prime}(c)=0 \Leftrightarrow 4 c-4=0 \Leftrightarrow c=1 . c=1$ is in the interval $(-1,3),$ so 1 satisfies theconclusion of Rolle's Theorem.

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Okay, So this questions asking us to verify the front that the function after practical to x square manage for class five satisfies two three hypothesis of roll student on the given interval Andan We have to find the number of sees that satisfies the conclusions of roast him. So here we know that after Max tickles too X square minus four plus five. So just by looking at this function, we know it is a poly normal because it has a a function of degree greater than one So here too So we know this is appalling amore and all polynomial czar continuous and they're also all defensible. That is sick That is the property of all polynomial. So we know off the back that thiss thes two conditions continuous and defense ability are are given of upon your meal And if you actually plug in the values of negative ones So if you plug in aa two of negative one squared minus four times negative one plus five, you get yesterday get eleven And if you do the same thing for three so now you plug in three the two times to re squared minus four times three plus five. It also gives you eleven. So we know that half of negative one and efforts three are also satisfied is also a truth statement. So now that we satisfied all the three conditions required for a roasted um, we can go ahead and find the numbers of sees that satisfies the astral conclusion. And so the recall the conclusion of this serum is that they that we know that there will be a number see, in the interval Judge that the director of the derivative at point C is equal to zero. So we have a horizontal line and we'll get you salt for this in this case. So if we take the duty of effort back so the derivative of F Prime I mean F Prime Vex, which is just a derivative. We can find that the derivative of effort back is just for eggs minus for and all they're doing is looking for where this function in the Cordillera. So all we do is set this function equal to zero. Then we just bring the fourth to the other side. So we get four x equal four on DH X equals one in self Rex and This is also our number, See, because if you plug in this value one, the four times one is one I'm in for ten points. Four minus four zero. So that is the sea that satisfied the conclusion? Of course, dear. Hey, there's our answer.

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