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Problem 24 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) = | x | $

Answer

The function $f(x)=|x|$ has a local and absolute minimum at $x=0$. It does not have either local or absolute maximum in its domain (which is the set of real numbers).

Discussion

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Video Transcript

we will sketch the graph of the function absolute value of X by hand. And use that graph to find the absolute and local maximum and minimum values of the function you need some in which in this case is the real numbers. So we know that the uh absolute value function is for the positives is just the same. So here mhm. So we have this is the axis. Yeah X and y. Yeah, we know that the positive the absolute part is the same value. That is the identity for the positive numbers. So we have something like test, let's say let's say this. Mhm. Okay. This man here a little bit. Yeah yeah At zero we have zero values zero because the actual value of 00 and we have this line and the important thing to notice uh to this line is that we have at one here for example The images one Olsen a two the image is too and so on. That is the absolute value is the same value. F X is positive or zero. And if X is negative we have yes rebellious, negative X. That is we change sign to become to have again a positive result. That is at one We have the same image and everyone were saying that one as well as for the number one to a native to have image to and so on. So we have the reflection of this line with respect to the y axis. Mm let's save this. Okay, so your line would be something like this. Yeah it's sitting here. So we have uh some some like these let's say Come on like 10 occasion mm And these lines uh extended infinitely to the right into the left, doesn't end. So if we take the whole domain of the function that is the real numbers. Mhm. We have only one absolute minimum. And in fact this is also local minimum at zero. So F has a local and absolute yeah minimum value At X equals zero. It's the lowest uh point in the graph. And for that reason is the absolute minimum. And also if you take any interval around zero we have that the image zero there is zero is the smallest value of the function. So it's also this both local and absolutely. But there is no local or absolute maximum. Has known local. No absolute myself. And that's the behavior is a function after the body of X. In its domain the real numbers. That is it has local and absolute minimum at zero. And they Value is zero. There is a local absolute minimum zero at zero at X equals zero. And no local nor absolute maximum value

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