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# Use the guidelines of this section to sketch the curve.$y = x^{5/3} - 5x^{2/3}$

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for the domain. We know that X is all real values, intercepts and free substitutes to your own place of X. We see that why is zero so 00 and we can factor this out and get X to the 2/3 x minus five equals zero. So he was sitting, Why equals zero? And you see the X can be zero. We already have that or five so five zeroes or other intercepts. And there is no symmetry to this graph and there are no ass in totes because it's not rational. It's not a rational function. Next for increasing and decreasing values, we find why prime, that's 5/3 X to the 2/3 minus turn over three X to the negative 1/3. So why prime if you factor this out 5/3 x to the negative 1/3 X minus two. So we see if we said our first derivative to zero, we see the X zero or two. So are credible critical points or zero or two if we make a line cracked for a prime, and we put our critical points and we test values around her critical points in the first derivative we see that we'll get positive values for less than zero between zero and two will get negative and anything greater than two positive. So it's increasing and then decreasing around zero. So this is a max local Max and we have a local men because it's decreasing in an increasing. So this is a local minimum value. Next, we're gonna test for Cohen Cavity. So if we already have our first derivative now we confined our second derivative. Why? Double prime is equal to sent over nine X to the negative 1/3 plus 10 over nine X to the negative four over three. We can factor this out. Well, I double prime equals 10 over nine x to the negative for over three X plus one. So here we have X equals zero. If we set this equal to zero, we have X equals zero or X equals negative one. So we make another line. But this time it's for F double prime and we're testing values around negative one and zero and our second derivative here we have negative positive negative. So the negative will give us con cave down positive con cave up negative con cave I'm sorry. This is positive. So this is Kong Cave Oppa's well, so here we see that there is a sign change, which means that at one there's an inflection point, so now we can graph it with the information that we have. So here we have intercepts 00 and 50 So 00 and let's call this 15 and we have a local men at two. So let's say this is too, and we have an inflection point at negative one. So it's called this negative one. And now let's find the Y values. So if we see where f of two is for our minimum value, we can say it's approximately negative 4.8. So this is negative 4.8, and we can say that's where our local minimum value is. Our maximum value is at 00 Where do you know that's an intercept? And we have an inflection point at one so here at a negative one, So a negative one it's going from concave down to Khan gave up. So conk it down, and the second we're a negative one. It's switches as an inflection to negative to. Khan gave up. And now our intercept 000 between 00 and two. Negative. 4.8. We see that it goes concave down. So it's rock on cave down. And then when it goes back up to our intercept, anything beyond two is gonna be calm. Keep up. So it's gonna be Khan gave up, hit the on the intercepts and and go No, and this is their growth.

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