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Problem 69 Easy Difficulty

Discuss the extreme-value behavior of the function $f(x)=$ $x \sin (1 / x), x \neq 0 .$ How many critical points does this function have? Where are they located on the $x$ -axis? Does $f$ have an absolute minimum? An absolute maximum? (See Exercise 49 in Section $2.3 .$ .

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{'transcript': "I don't talk about question of 67. So we have to find home in the critical points are the What is this location on access is whether it has absolute minimum maximum. So for all that, we have to differentiate FX. So Russia is. We have to use the production here. So next time you have to use the change earlier for the transition of Sinus cause. Then we go inside and the transitional bottom or excess minus one of our excess square. Then we have the full of the second term, as it is, and then we differentiate the first time, which is X, which is just one. So we have X over extra squares just minus one or x times calls of one or X plus sign of one or X. This is the value of flashbacks. In order to find a critical value, we quit this 20 so we have F Dash X is equal to zero. This means that minus one over x cause of one over X plus sign of one of our experts, including zero. This means start sign off. One fine sign of one over X is equal to one over x Uhh! This is not Dan. It's cause So this is one of our X cause of one works can also better than us. Dan, off one works is equal to one over X. This is how what we're getting, uh, in order to simplify this, let's if he assumed one over X is, let's say 80. That means that we're getting Dante is equal to t. Now, this particular equation is basically the intersection between the curve off Dante or let's say why is equal to 20 on Why is equal to see what this means is if we draw just one period off. Dan, it is like this on this is over. This is by what? When this aspire what minus by what went via Quito, Michael rupees a straight language Schools like this. So since that tan is going toe, tell the infinity So definitely this means that the points of intersection will also be in finite. So we'll say that over here we are getting in finite critical points on on those points. Definitely We will be having, uh, local minima as well as well have local maxima on as far as the absolute minimum or maximum concern since over here will have if extends to plus all minus infinity. Then we have the value off FX as it will be plus or minus infinity times Sign off zero which is just we hear that will be zero. So when we're using the limits in that case will come out as let's use the limited that case limit extending towards plus or minus infinity way. Have limiters x times Sign off one works. This can be read it on us limit. Let's say zis tending towards zero and this can be returned a sign, Z or Z and we know that this value is actually one. So what this means is it has an absolute Maxima off X equal to offer one. This is the absolute Maxima among these local minima among these local minima, it will also have a nab salute minimum as well. Since one off, these has to be the lowest among every point. So that will be absolute minima On the location on the X axis will be all those points which is satisfying this. So if we talk about one solution that is speak with zero. If we talk about another solution That will be the point of intersection off Dante. Andi. So we have to solve this graphically, so we'll get infinite solutions in that case. So that's how this X times kind of wanna works will be."}