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# (a) Sketch the graph of a function that has a local maximum of $2$ and is differentiable at $2$.(b) Sketch the graph of a function that has a local maximum of $2$ and is continuous but not differentiable at $2$.(c) Sketch the graph of a function that has a local maximum of $2$ and is not continuous at $2$.

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We have three parts in this problem for eight. We sketch the graph of a function that has a local maximum at two And is differential at two Barbie. We sketch the graph of function that has local maximum at two and is continuous but not deferential differential at that point. And in pursuance sketch the graph of function that has local maximum at two and it's not continuous at that point. So the story part a the graph of the function that has a local maximum at two and its differential to here. By the firm a theorem, we know that the derivative that exists because the function is differential a lot to get to be equal to zero because we have a local maximum. So this is a sketch of a graphic function that has a local maximum at two. And as we can see being differential a lot too. The graph has a recent all tension line at this point of transparency, which of course or where of course the uh local maximum. So by furman theorem, Yeah. If Derivative to get to be equal to zero. That's because we know we are saying that The function has a local maximum at two and its differential there. So then we apply for material, we're not ready to be serious. So this is this is an example of that. We have a local maximum value at two. Okay, party, we want a graph of a function that has a local maximum. Two again, but now the function is continuous but not differentiable too. And this is an example of that. Again, we have a local maximum at two because if you look near the valley too, that if we take an open interval containing two, the highest values of the graph he seemed to in this case. And so we have a local maximum two. But the function he's not let's call function F is not differentiable A two. Because we have this corner here at this point that says that there is no tangent line and F is continues. There is continuity all in all points of the maid of this function. So we have the three properties. So we have a graph where the change in line is zero. That is there is derivative and must be zero because there is a local maximum at the point. And here we have a local maximum but And continuity but we have no derivative at the point. And parsi, we want a graph of the function that has local maximum two again and it's not continuous. And that's Uh huh. In an interesting case because these uh barbie, it's almost a solution. But we get to take the image of two away from the trend of the lines. That is, we put it a little bit up this open circle here Means the image of the point is only at the .2 is put another place in this case is put at this. Hi here, it's only for one point. But graphically we make this type of draw in order to indicate that. So we have a function that is, this continues to the images. Put a little bit up the image, attitudes. Put a little bit up in such a way that if we take any interval containing to the highest uh huh value of the function is the image to in fact, so it has a local maximum, Yes. Local maximum two and uh is not continuous at that point. And of course not differential rules. So we have these three solutions today given problem.

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