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Problem 8 Easy Difficulty

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 4 $, absolute minimum at $ 5 $, local maximum at $ 2 $, local minimum at $ 3 $

Answer

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Video Transcript

We want to sketch the graph of a function that is continuous on the interval 1 5 And has an absolute maximum at maximum at four. An absolute minimum at five. Local maximum at two in the local minimum at three. So we have a sketchy grant with those properties. Let's see The absolute maximum of course at four. First of all we see that the graphs continuous function. There is only one piece with no job or discontinuities. So it's a continuous function and it's a fine on the interval 15. Then we have an absolute um Maximum at four. We can see that this value here is the highest point on the whole domain of the function. The whole graph. So that's the absolute maximum in this case is also a local maximum. We have local maximum, Sorry, an absolute minimum at five. And we see here that the lowest point in the graph and the whole domain is This family here at five. We here we have an absolute any mom and then we have an absolute uh sorry, a local Minimum of three. Also local minimum three. We have or the only in this case the only local minimum Because if we see close to Valencia three, We know that the image of that .3 is the smallest value of the graph around these value three. But it's not the lowest the value of the whole graph because they're all lawless violence. They're sort of minimum is attained at five. But this is a local minimum of three. This part here now we have a local maximum up to so to have a local maximum here because if we look around two very near the value to, we have the highest point Or value in the graph corresponds to the image at two. See we have a local maximum and here we have a local minimum. That's it. We don't have any requirement at one. The important thing is that one, we have an image that is not and that is neither the absolute minimum nor the absolute maximum in order that The stream values are obtained at four and 5. And that's an example of the function fulfilling this for robert is here we can remark that two, we have an absolute maximum, but also in local maximum and that's it.

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