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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) = \ln x $, $ [1, 4] $

$f(x)=\ln x,[1,4] . \quad f$ is continuous and differentiable on $(0, \infty),$ so $f$ is continuous on [1,4] and differentiable on (1,4)

$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \Leftrightarrow \frac{1}{c}=\frac{f(4)-f(1)}{4-1}=\frac{\ln 4-0}{3}=\frac{\ln 4}{3} \Leftrightarrow c=\frac{3}{\ln 4} \approx 2.16,$ which is in (1,4)

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Okay, so we're now being asked to verify that the function satisfies the hypothesis of the main very term under given inedible. And then we're being asked to find all number sees that satisfies the conclusion of the main writer. So our function in this case after backed you called Alan Bax and the call at the graph of f of X of Ellen of X is actually look something like this. This is where X equals one. And it is important to note that Allen of X is not is only defined for numbers greater than zero. So its domain, it is really ex greater than zero, because at zero, it goes after negative. Infinity, however, were only being asked to define from one to four. So which is a continuous function. And by looking at this graph of Alan events, we also know that is defensible because there you can take a tangent line at any point. So we know these two conditions are satisfied, which are the conditions for the mean value there, Um, and the mean very serum staged that there is a sea just at the derivative. At that point, C will give us the average slope, which is F of four minus one. What's that one all over a four month one could. That tells us the average folk from one to four. And we can ask you find the numbers. See, that satisfies this condition by directly clock in this equation. So the derivative of Ellen of Axes one of Rex, But we're going to be writing in terms of sea. So that's why only one oversee and this is equal to Alan for minus Alan one all over four minus one. And since Ellen of one is zero, as you can see in the graph, this is simply zero. We get one over C is equal to Ellen for over three. And then now we can solve for C by just reversing. Just flipping this equation. So we got sea Eagles totally over island for So this is the answer. This value of C well, give us inhabits slope. So just for a visual reference, what this is telling us is that some assume that this is point for that the average slope from one to fourth of this slope right here there's a value three over Alan for was specific, which will give us the exact same slow at as it is from here to here. The tangent on this graph at this point will give us the slime flow. Let's just say it's like it looks like it's right here, right here. That's parallel to it. This is three over Ellen, for that's all the major that there was telling us.