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Okay, so for problem 47 we have the function
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a shacks which is defined by X cubed, divided
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by three minus two X squared and plus four X
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Where domain iss from zero to infinity. Okay,
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now you are a We need to take the derivative
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of dysfunction. We have X squared minus four acts
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us four, and we let this to be zero
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. Then this is it. Equivalent to X minus
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two squared, which is equal to zero. And
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our route will be too. And so, h
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two, we'll be a loco extreme. Now,
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is this a, um next thing to do is
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to identify whether this value is is an absolute extreme
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. Um, the way to do that, just
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a first. We need to use directive and let
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the dream, too, to be bigger than zero
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and find out the the increasing interval. So seems
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this expression can be transforming to exploit it to where
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, and this is always an elective on. So
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that means our increase interval will be from 0 to
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2 union to to infinity. We have to ethics
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have to exclude to in this case, because when
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X is equal to two, then our expression will
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be zero and that it is not, um,
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bigger than zero anymore. So that means our function
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will be like this. It's from 0 to 2
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and two to infinity, so it is always increasing
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. It's kind of like that should be like this
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and nothing too. Uh, another thing to notice
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. Instead, the function here has the local extreme
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, but it is not. It is not an
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absolute extreme, because the function is it's increasing from
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two to infinity and well goes to infinity. So
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there's no upper bound and sing reason. There is
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no there were bound to this, uh to dysfunction
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or the minimum value here. The the minimum value
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for or dysfunction on this on the interval from zero
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to infinity is at zero and so two. It's
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not an absolute extreme. So that is our conclusion
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up to is not absolute.