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University of North Texas

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Problem 12 Medium Difficulty

(a) Graph the function
(b) Use l'Hospital's Rule to explain the behavior as $ x \to 0 $.
(c) Estimate the minimum value and intervals of concavity.
Then use calculus to find the exact values.

$ f(x) = xe^{1/x} $

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Video Transcript

we want to first the function EPA Becks is equal to times E to the one Rex. Then we want to use locals rule to explain the behavior as X on purpose. We also want to estimate the minimum value and intervals of calm cavity. Then we want to use how close to find those values. Exactly. No, I went ahead and Ari graft this function here. And so now, looking at the graph, you can see that to the right and to the left of zero. We approach different values, cause to the right of zero. Looks like he's going to positivity. And to the left of zero, it looks like it's approaching zero. So for part B, we're going to need to do to interval to limits one from the left and one from the right. So let's just go ahead. And is that so? The limit ads ex purchase a long ago headed deport. Be so limit as X approaches zero from the right. So if we have X times e to the one over X, if we were to go ahead and just plug in zero well from the right zero R excuse to zero and then one of her ex would go to positive infinity and e to the positive. Infinity is going to, um, go to infinity. So let's go ahead and re brightness as the limit as I approach zero of the right Oh, heed to the one over X over one over X. So now if we were to go ahead and apply the limit, the new Margo sent Fizzy and our development goes to infinity. So this is in a form of which we can apply. Look tells you, let's go ahead and do that. So this is what equal by local calls too, the limit as X subscript zero from the right of so to take the derivative e to the war breaks we're gonna use changeable. So we take the derivative of outside which will not change. And then we're going to have the derivative with respect to X of one uber X. And you might notice that we also have the derivative of one of our excellent denominator. So just cause I'm lazy, I'm going to go ahead and counsel those out as opposed to taking the derivative of them. And doing that will leave us with the limit as X zero from the right of E to the one of Rex. And now that's going to be e to something approaching infinity. So this will diverge to positive infinity. Now for the next step, the limit. As this approaches from the right, we can apply the same logic again. And we're going to have the limit of X approaches, zero from the left. And so, actually, let's first go ahead and see what we get when we do this. So the limit as X approaches zero from the left is going to go to zero end limit as e are so first limit of one of her ex is going to do with the negative infinity. So that diverges. But then e to the negative Infinity approaches zero. So I actually have zero time zero, So it goes to zero. So the only thing we actually needed to use loopholes or was the first part of the question All right, now for part C, we want to find our minimum, so I can only assume in this they meant a local minimum as a opposed to a true minimum sense. Till elect dysfunction goes to negative infinity And so when I looked at my graphing calculator, it looked like the minimal occurred at one two points or one for X value and 2.718 for our wide value. And now let's go ahead and look at our intervals of con cavity. So it looks like to the left of zero the functions just calm, keep down. And to the right of zero, the pictures can't give up. So conned cave down. Well, that just looks like it's gonna be from negative energy to zero and or con que up. That looks like zero to infinity. So maybe we chose good intervals here. Or maybe we should have zoomed out a little bit more. So now we actually want to use calculus too. Find this. Actually, let's go ahead and check it out. So no of ex again waas x times e to the one bricks To take the derivative of this where I need to use product So we leave the 1st 1 the same, take the derivative of second and then we add them in the opposite yours e to the one of rex times the derivative of X. So first, the derivative of X is just going to be one. And then to take the derivative e to the one of Rex. Well, we already did that on the first round. We said it was going to the E to the one of rex times the dodo one over X, which should be negative one over x squared so we can go in and simplify this. Now, as so, these X is in the first and only put twenties around that will cancel oversimplified to e to the one of rex over X plus e to the one of rex plus one. So I'm gonna factor plus one times one. Now, I'm gonna go ahead and factor out the E to the one brooks and then just kind of rearrange it to be won over one minus X. So we know this here always be strictly larger than zero. So if we want f prime of her f double prime movements, yes, this should be a but, um, we're looking for the first derivative right now. I'm getting ahead of myself. So to find that minimum, we want to set this equal to zero. Now. I'm just like I was saying, though this here should always be strictly greater than zero. So we just set one minus one over X equal to zero. And doing that, we'll get one minus one over X equals zero. Add the one over X over. We'll get one is equal to one over x reciprocating side, and that tells me one is just equal to X. So at least X value we got before waas the safe and now we can go ahead and plug that in to see what our minimum value should be. And so that is going to be one e two, the one over one which just isn't being e. And that is approximately 2.718 So that one checks out. Now let's go ahead and book for intervals of conch empty. So we had F Prime of X is equal to e to the one of Rex one minus one of Rex. All right, now, when we take the second derivative or enough to use product will once again. So it's gonna be easy to the one of Rex times, the derivative with respect to X of one minus one over X and then plus them with the opposite ours. One last one over X times the derivative of age of the waterworks so derivative of 10 derivative of negative one over X will be one over X squared and again the derivative of he did. One of your ex is going to be e to the one of Rex times negative one over x squared. So he's gonna be over X squared plus one minus one of her ex times e to the one of her ex minus one over X squared. And now let's go ahead and factor out this e to the one of her excellent distribute multiplied as well as the one over X squared. So it should be easy to the one over X over expert. So the first part will become one then plus one minus one over X and on. I forgot to distribute this negative right here. So distributing that negative inside here would actually switched the signs. Right? Glad I caught that. Now the ones cancel out and we're just left with e to the one of rex over x Weird times one over X or there. Yet we can write that as execute now again, like I was saying that last part are numerous. Riker will always be greater than or equal to zero. So wherever denominator will be negative, impossible. Decide our interval. So to the left of zero X cubed is negative and to the right of zero, X cube is positive. So we just get f double Prime of X is so Kong came up con cave up on zero to infinity. And remember, this is one F double prime is strictly larger than zero and we're going to have that function. Is Khan Kate down on negative Infinity to zero, which is when f double prime is strictly less than zero. So it looks like our graph this time really did give us all the information we really needed for this. And we really didn't need to estimate it or use Coco's to really get exact about use.

University of North Texas
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