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Draw the graph of a function that is continuous on $ [0, 8] $ where $ f(0) = 1 $ and $ f(8) = 4 $ and that does not satisfy the conclusion of the Mean Value Theorem on $ [0, 8] $.

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Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Okay. The question here is asking us to draw the graph of function after this continuous on the close interval from zero to eight where F zero is equal to one and every great is equal to four. And that this graft does not satisfy the conclusion of the Minbari term on the close interval from two to eight. So let's just recall the conditions required for the But I mean I assume so. The first one is continuity, so it has to be continuous on the closed in a room from zero to eight. So this is the animal, and the second one is defense ability. So you can take the dirt of everywhere on the point on the open interval from zero age. But it does not include the endpoints. So here they just asked us to draw a graph of something that does not satisfy these conditions. So we know that this one does satisfy continuity because we can draw this function without having to pick up a pencil, or and we also know that the limit out all these points exists. But that's another way to find continuity. And however it is not defensible at every point because we have a sharp corner here and a sharp corner means air. Infinitely many different possible derivatives. So different possible tangent line that can occur. You're so this's not satisfy the condition for roasting him. However, what does it mean when one of these fails? It means that this condition right here, if that s ocean, see what I mean Value themselves is that these conditions were satisfied. We will know that they're there would exist in number see in this interval from Syria eight such that and half of eight minus F zero all over eight miners do what exists so that there's a pointer is an average slope. So this is telling us that there's a slope from this point. So the average slope from here to here that there's a there's a tangent line there was a derivative. There's some derivative or tension line that exists on this function that equals the slogan of the But since we failed six this Sims, this graph has failed this condition. There's no point, absolutely no problem. No numbers sees no function X, no number on the X axis that has that will give us attention. Line that has descend. Exactly, because if we try to do that, look, if you obviously here there's a tangent line this way. The tangent line is this way. There's flow disrupted, not equal zero. And we can't take the derivative here because this is a sharp corner. So there's no actual tangent line that has the exact slope from here to here. So this is why this graph fails the conclusion of the mean value there.