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Draw the graph of a function defined on $ [0, 8] $ such that $ f(0) = f(8) = 3 $ and the function does not satisfy the conclusion on Rolle's Theorem on $ [0, 8] $.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 2

The Mean Value Theorem

Derivatives

Differentiation

Volume

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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okay. The questions I think of to draw the graph of function defined on zero a just at F zero is equal. Effort way is equal to three and the function does not satisfy the conclusion of rules term Conroe Systems on the interval from close interval homes. Ota So here the anticancer drug a function that does not satisfied with the conclusions of rostrum. So if you recall ah, the conditions for Rose dirham are continuity continuity on the closed in a roll so from zero to a and the second condition is different ability. So it has to be defensible. You can take the derivative at any point on the open interval, which means it does not include the endpoint. And a third condition is the end points of the to function at the function equally says so, the beginning point and the endpoint equal each other. In this case, it would be F zero enough of those to equal each other. So the questions asked us to draw such a function where this condition is true but does not satisfy so one of these will not be right. So a very common example is dysfunction and say Ah, that is a horrible teacher. I couldn't finish to that again. Okay, um, this this is a straight line, but so essentially this function, it is continuous. On this interval, Aiken, draw this function without picking up my pens or my mouth in this case. So this is true. And we know that the end points after zero, which is three and after eight, which is also three is Xu said this is also true statement. But this conditioner here defensible on jury, this isn't true, because at this point, whatever point this, maybe Let's just call it eggs. Um, this point at acts on some normal between zero and eight. I can't take the derivative here because this is a sharp point, and you're not allowed to take a derivative at a sharp point because they're infinitely many possible derivative. Because if you look closely at a sharp point as I've grown here, you can take you can have a derivative like this. You could have a change it like that. You could have attended like that. Every Tanja like that, and it's just inconclusive. So this statement right here is not true. So this is a graph of function that is not satisfied. Conclusion ruled him. And if we take it further, if we remember the conclusion of the rules to him, it's so that there will be a sea in the end. In this interval shut that the derivative is equal to zero and you can see in this funk and there's no point where there's a derivative way. If we take the tangent line at at any point in this function, it would be equal to zero, because here you just have a slope. You have a slope here and then at this point, as I said here does this is not a place where you can differentiate, So this is a graph that just not satisfied the conclusion of rules to him.

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