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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.$f(x)=5-12 x+3 x^{2}, \quad[1,3]$

$c=2$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 2

The Mean Value Theorem

Derivatives

Differentiation

Applications of the Derivative

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the function. Fearful Wrecks the Eagles five minus 12 X plus three x squared on interval from X equals +12 X equals degree. So this was the first to have about this use of rules there, Um, because it is a quadratic function are second order pulling over your function. Eso By that functional definition, it is both continuous and differential on the required intervals. So now we need to check for the third hypothesis of Rose therm, which is that the function is of equal value at the end points of integral. So first truck, the function at X equals one. We have fired more honest toil times one post three times ones squared, which is equal to minus four. Um, a function at X equals three is five minus 12 times three plus three times three squared, which is he also equal to minus four. So, no, we should be able to find at least one value between one in three. We're most slope at that point. C is zero. That is, we need to be able to find at least one value where the salt might change from positive to negative or vice versa, or if the function happened to be constant for the entire range, which will not be the case here. Then it would be the entire inch, but we're here to find that out. So let's take the derivative of the function every edible, minus 12 plus three times two x. Let's set that equal to zero. So then minus told host six X Eagles here, move over. A 12 do right, right. Six were you result in having a see value of two, which

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