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Numerade Educator

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Problem 29 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = x - 3x^{1/3} $

Answer

see solution

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Video Transcript

So here we can see that the domain is X is all real. And for the intercepts, we If we factor this out, we have X to the 1/3 times X to the 2/3 minus three equals zero. So we have X equals zero. So we got 00 and we also have X to the two over three equals three. And that's the same thing as X to the X equals three to the third to the power of 1/2. So we have X is equal to plus or minus three square three. So our second intercept is plus or minus three square a three zero. And for our symmetry, we can, um, plug in one a negative one for and easy way to tell what it is. So we do one minus into this function one minus three times 12 The 1/3 that gives us negative, too. And we plug in negative one minus three to the negative times, negative one to the 1/3. And that's positive, too. So they're the same number, but opposite signs. So this is odd. And for Assam totes none because it's not rational. And now to find the increasing and decreasing values we find the first derivative. So the first derivative is why equals y prime equals one minus X to the negative 2/3. And if he said that zero to find the critical points we have one over X to the power of two over three equals one or one equals X to the power of two over three. So X is plus or minus one. So that's our critical points and what's also find the second derivative over at it. So why Double Prime is equal to to over three x to the power of negative five over three. And if he said that to zero, it looks like two over three snaking bigger to over three X to the five over three. All in the denominator equal zero and we see that why Double prime is undefined at this point at zero. So we're going to say it's an inflection point, all right? And now if we draw our line and we put negative one critical point and then also one and then we put our inflection 10.0 So it's this is the inflection 0.0, and this is we're testing of Prime. We're doing the first derivative test, and we can see that this is positive. Here. Negative Here. Negative here and positive here. So from positive to negative. Increasing, decreasing. So this is a max. And here we have from since it's on negative to positive. This is a minute. So here we have vermin. This is the first riveted test, and also we can see what the Y values are for those F one and F negative one. Here we have negative too. And here we have to, like we stated earlier with the symmetry and F zero at her inflection point zero. And we know that's an intercept. So now and we also know that why double prime at one is greater than zero. So that's his con cave up. And why double prime at negative one is less than zero says Khan came down and now we can graph it. So we know we haven't intercept at negative in positive three square of three and 00 So we haven't intercept here, here, in here. And now we see that at negative one. We have negative 12 We have Hey, Max. So this is too And let's say this is negative too. So here we have a max. Uh, let's say this is negative one, and this is positive one. My bed. Negative one positive one. And here we have a max and positive one. Negative, too. Is it men? And we see that it's Kong cave up here. Here's our inflection point and its Khan came down here and this is our graph.