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Numerade Educator

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Problem 42 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = (1 - x)e^x $

Answer

Because of the change in concavity the inflection point is:
$(-1, f(-1))=\left(-1, \frac{2}{e}\right) \approx(-1,0.736)$

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Video Transcript

domain for this function is X is all real. The intercepts are 01 and 10 There is no symmetry. And for Assam, toots, um, there's no vertical, but we can test for horizontal. There is one at, um negative infinity. So let's let's test as X approaches negative infinity for a function one minus x to the, um times e to the X. And this limit will give us because it's negative Infinity, It's going to give us zero. So as X approaches negative infinity or why value is going to approach zero. All right, so that is our esteemed too. And now we can look for increasing and decreasing intervals. So why prime? Why? Prime is equal to negative x you to the ex. And if we set our first derivative to zero, we can see that a critical 00.0. So if we do the first derivative test around zero of print, so it's positive and this interval negative and this interval, So that means that zero is going from increasing to decreasing. So at zero, we have a local max and you know that 01 is our point. So 01 is our local necks. So that's it for the first derivative test. And now we can take the second derivative and find the cone cavity of her graph. So why Double prime is equal to negative X minus one times E to the power of X. And if we set that equal to zero, we see a critical point at negative one. So the second derivative test for Con Cavity around negative one. We can see it's pot. It's Kong cave up here and it's Kong cave down here so can't give up and con cave down. And if we test where f of negative one is, we see that it's 0.7 positive 0.7, all right, And also because this is changing from positive to negative. There's a sign change so that this is an inflection point. Okay, so that's it for the second derivative test, and now we can graft. So we haven't asked himto. Let's grab that first in green. We haven't asked him to at like, Well, zero and we have a new intercept at 01 So it's say this is one. All right, this is one and say this is negative one. All right, So now we have an intercept at 01 and 10 and we know that our graph as approaches negative infinity. So in this direction, it's going to go to zero the line zero. So the more is over here, it's gonna approach zero. But let's see what's going on here. So since this is a max, it's going to be increasing before zero. But after zero, it's gonna decrease. All right, so it's gonna decrease here after zero, and it's gonna be calling cave down. Gonna hit the access and become came down. And then on the other side, it's going to be con cave up. Um, after it hits the inflection point. So about negative 10.7. So let's say that's about here. This is our inflection point. So it's gonna be con cave up and increasing and then con cave now in here. This is a graph