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Numerade Educator

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Problem 46 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = e^{2x} - e^x $

Answer

Graph will be concave up when $y^{\prime \prime}>0,$ that is when $x \in\left(\ln \frac{1}{4}, \infty\right)$
Graph will be concave down when $y^{\prime \prime}<0,$ that is when $x \in\left(-\infty, \ln \frac{1}{4}\right)$

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Video Transcript

for this function. We know that our domain is exes. Oh, really? Our intercepts. If we substitute zero into X, we can see 00 is an intercept for symmetry. There is none. Assam totes. There's no denominator here. So we can't set that equal to zero to find vertical. But for horizontal ask himto What we can do is set. We can find the limit as X approaches Negative infinity Because that would seem like it would find, um, an ass in tow for our function e to the two x minus. Need to the eggs substituting negative infinity into X into our ex ever function will give us zero. So we have a horizontal ask himto at why equals zero. And now we can look at the increasing and decreasing intervals by finding the first derivative. So why prime equals two e two the two x minus Easy the X All right. That is why prime. And if we set that equals zero, we confined critical points. So's factor this out. So that's equal to e to the X Times to e to the X minus. One equals zeroes. That's easier to see. So it's of the X equals zero. We can't do that. But for this one, we can see that X is equal to Ellen of the half, all right, and that's equivalent to Negative Ellen of two, which is approximately negative. 0.693 So that's a critical point. And if we make a little line for our first derivative test since his F prime and this is a critical point negative 0.693 every test values to the left and right of it so smaller than our critical point will give us negative and anything larger will give us a positive that puts. So if it's has a decreasing interval and then it goes to increasing, that means that this is our local men local minimum value here. And if we want to find the Y value of this minimum value of so substitute into our regular function into why so f of are critical point, which was negative ln of two, is the precise that will give us negative 1/4. All right, so that's it for the first derivative test. And now we can test for Kong cavity using our second derivative test. So if this is Ah, Why prime we can solve for y double prime, which is e to the ex 24 e to the power of X minus one. And if we set that equal to zero, then we see that are critical. Point is negative too, ln of two, which is about negative 1.39 So if we do the second derivative test, so this is F double prime. We can see that, um, and we put our value here. So negative 1.39 Anything smaller will give us negative, and anything larger will give us positive. That means that this area is Kong cave down. This area is Kong Cave. And because it's switching signs from negative to positive, we can say that this negative 1.39 is an inflection point. And if you want to find the why value for this point, the exact as we already said, that was negative too, Ellen of two. So plug that into our function. Why, That will give us negative three over 16. All right, so this is our second derivative test, This tests for Akane cavity, and now we can graft or function. All right, so we can test, we can plot our minimum value first. We said that waas negative ln of to which we said waas we said that was approximately, um, negative 0.693 So that would be if this is one, it would be about here. So let's say this is negative. Ellen of two. This is negative one. So that's negative. Ellen of two and negative 1/4. So this is one. If this is negative one for the wise, it would be about here. So negative 0.25. So we can say this is our minimum says their minimum value in her inflection point was negative to Ellen of two, and that was approximately negative. 1.39. So one point negative, 1.39 would be about here. And the Y value, he said, was negative. Three over 16. So let's say that's about here. Let's say your inflection point is about here. So that's our inflection point. Um, when we said that we had ah, horizontal ass until at why equals zero since rather than all right. And now we can con cave down before negative 1.39. So before our inflection point. It's Colin Cave down, and then it hits our inflection point. And now it's going to be con cave up until it hits our minimum value. And now it's going to continue being. Khan came up beyond that as well, and it's gonna hit. Oh, we have an intercept here. Don't forget 00 So minimum value, it's gonna go. Khan gave up and continue. This is our curve.