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(a) Use a graph of $ f $ to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.

(b) use a graph of $ f" $ to give better estimates.

$ f(x) = (x - 1)^2 (x + 1)^3 $

a) concave up: $(-1,-0.3),(0.7, \infty)$

concave down: $(-\infty,-1),(-0.3,0.7)$

inflection points: $-1,-0.3,0.7$

b) Inflection occurs at $x \approx-1,-0.29,0.69$

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Oregon State University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Okay, so we're being asked to use the graph of F Forgive a rough estimate of thie interrupt, contact bree in accordance appointed inflection. So let's go ahead and start. And so contact when it is cranking up when it has that sort of u shape to the function in this case, dislike hers. You could see this u shaped right here on occurred from negative one to some number right here. We can say negative zero point two. This is only a rough estimate, and they're gonna want to negative zero point two approximately. And then we see it again somewhere over here, but a little bit closer to one. So this is going to be something like point seven. So you see this kind of shape, but it goes forever. So will be one seven to infinity and then concave down the upside down u shape. You could see that between you could see that between negative zero point two and point seven, and we can also see it right here. This's a little bit harder to tell that this is a upside down you shape. So from negative infinity to negative one so negative and serenity, negative one. So that means the points of inflection occurs when the sign children does the crews. That negative one. Um negative. Teo point two. And at zero point seven. And those are rough estimates. And to find we're going to be using a graph of double prime to get us a better estimate. Ah, so we take the graph of F double prime and after we'LL try back to his vehicle. Ah, if you take the derivative again, you get twenty x cube, close twelve X squared, minus twelve X minus four. And we're looking for when this is equal to zero. So to give us a better estimate of how where the con cavities are and I took delivery of drawing out thes of function And as you can see, our approximation wasn't that bad. We had negative one zero, and then it is at negative. They're going to nine. Very good. And in point six nine. So we didn't have that bad of an approximation, so I could have, and that's Ah, this is when it crosses. So this is going from negative to positive. So this is negative. His Khan came down the continent down from negative infinity to negative one, just in case you did not read it. And if it's positive this conclave up, so anything above anything after negative one zero and it changes again. And negative zero point two nine, which we estimated again, disaster closure, important threes. But we have Point Teo that bad either. And then it changes again. At point six nine, we said point seven, which isn't as horrible as well in our inflection points are also correct, so that is it.