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Numerade Educator

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Problem 17 Easy Difficulty

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1/x $, $ x \geqslant 1 $

Answer

Absolute maximum $f(1)=1$.
No local maximum. No absolute or local minimum.

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Video Transcript

here we are going to sketch the graph of the function F of x equal one over X. For X greater than or equal to one. And is that raft to find the absolute and local maximum and minimum values of the function. So here we have made a sketch of the graph. We know that if that one is one At two is 1 half, At three is 1/3 and so on. So the value of the function is decreasing or not. My anatomically and it never stopped decreasing. That is always increasing. It means the graph is always approaching the X axis or is getting closer to it as we go to the right but you never cross it. Or that is it never gets a value that is on the X axis. But it's always approaching or getting closer to. That means that limit when X grows without any bound. That is what eggs goes to plus infinity is zero. But coming from positive values, that is the function is always positive. That's the behavior of dysfunction. And we can see clearly that the highest point in the graph in this case, that is included in the graph is F of one which is equal to one. So if of one equal one is the absolute maximum value of the function. That is because we have considered function only for X rather than or equal to one for A Square. That articles one. And that is the sort of maximum values one and is obtained in fact at X Equal one. So we have this result, we have no local maximum for this function remember we can consider this same value as local maximum because we have we don't have any graph to the left of the functions. So we only consider local streams inside of the main the endpoints of the interval. So in this case there is no local maximum, right? And there is no absolute minimum either has no absolute minimum or local minimum and that's because there is no lowest point in the graph, that is the graph is all with descending. It's never never stopped descending, so there is no longer with wine. The thing is that it is descendant but in a way that it will never touch the exact size of the descending, never stopped. And for that reason there is not the lowest point in the graph. And so there is no absolute medium and for similar situation there is no local minimum, so we don't have either of those. And in summary the function has only an absolute maximum value of one Attain or the course at execute one. And that the characteristics off or the properties of the function F Mexico one over X for X prison records one