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Numerade Educator



Problem 51 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = xe{-1/x} $


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Video Transcript

Hey, it's Claire is the one you right here. So we have f of X is equal to X e to the negative one over acts. So the domain is all real numbers except zero. Since he can't do division by zero, you know that F zero is undefined. So there's no why intercept And we're gonna test the f A negative X when we get negative x b, that one over X which is not equal to F acts which is not equal to negative off of X. So we see that it's not an order, even function. And then we look at limits. So the limit as X approaches infinity for the function of eggs we get infinity and for the limit as X approaches negative infinity for the function began Negative infinity for the limit. As X approaches the positive side of zero, we have zero. And as the limit approaches the negative side of zero, we see that it becomes zero negative zero times infinity but just equal to negative and friendly. So as you go from, look at the right side approaches the origin and then the left side of zero approaches the vertical ascent tote, which is equal to zero, is equal to X. We're gonna find where the first derivative is equal to zero, which is equal to eat the negative one over X one plus one over X, and we get Xs equal to negative one. It's also undefined at X is equal to zero, so we look at the intervals. Negative affinity for my negative one. Let's choose negative, too, when we get bigger than zero, so it's increasing from negative 1 to 0. Let's choose negative 0.5, which is decreasing since it's less than zero and from zero to infinity. Well, choose one, and it's increasing. So we see that there is a local max, a negative one comma negative. Next, we're gonna find where the second derivative is equal to zero, which is equal to one over X cubed E to the one over X. We see that it's undefined when X is equal to zero and I'll be negative when X is less than zero. F will be conch IV down, and it'll be positive when X is bigger than zero and it'll be calm. Keep up. So when we draw our graph, we have a hole in the origin. And I look something like this, and I don't look something like this. And we have this point would use negative one common negative. 2.7183