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Numerade Educator

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Problem 21 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 tranformations of Section 1.2 and 1.3).

$ f(x) = \sin x $, $ -\pi /2 \leqslant x \leqslant \pi /2 $

Answer

Absolute maximum
$f\left(\frac{\pi}{2}\right)=1$. No local maximum. Absolute minimum
$f\left(-\frac{\pi}{2}\right)=-1$. No local minimum.

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Video Transcript

we're going to sketch the graph of the function sine of X for eggs greater than or equal to negative by half and less than or equal to by half. That is defined on the closed interval from negative pi half pie, half. And and with that breath we are going to find the absolute and local maximum and minimum values of the function. So we have this access here. You know the function Sign at by half is equal to one In a negative biophysical 2 -1 and say he's here. Let's see here we have by half example and here we have negative playhouse for example. And more or less we have this graph here and here. So this is the sine function and we are including the end points of the interval that is by half is included. So it seemed much one is included. And the image of niantic by half is also included. And we can easily see here that we have an absolute maximum. Mhm Mhm. And the maximum value is absolute maximum value is the function over this interval is one and this value of course at X equal by health. That's because my house is included. Either the main there is the main is closed to underwrite in point And for that reason the highest point on the graph is the image of by half at this point here included in the craft. So is the absolute maximum value of one which of course by half. F has no local, no local maximum. Mhm Because inside the interval there is no point which has a his father near the point. And we respect to the lowest point in the grass. In fact, we have one. Is this one here? So F has an absolute minimum value. Okay, negative one. And the value of course add X equal negative by half. And that's because we have included the left hand point in the domain. So this lowest point existing this function and F has no local many of them because there is no point inside the domain inside the interval, not including the impacts are inside the interval death as the that be um the most value near the point. So we have these four things here at least. We have absolute maximum minimum. That's the maximum values one at by half. And the absolute minimum value is is uh native 1 90 by half. And there is no local extreme for this function. So that's c result.