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(a) Find the critical numbers of $ f(x) = x^4(x - 1)^3 $.(b) What does the Second Derivative Test tell you about the behavior of $ f $ at these critical numbers?(c) What does the First Derivative Test tell you?

a) Critical numbers: $x=0, \frac{4}{7}, 1$b) See explanationc) Ist derivative test tells us increasing/decreasing intervals, and that there is a local max at $x=0,$ local min at $x=\frac{4}{7}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

Missouri State University

Campbell University

Baylor University

University of Nottingham

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Okay, Tio, we're being asked to first find the critical number of the aftereffects of the way we do that. We take the derivative and we settled here for the zero. So in this case, we're going to apply the product. And, General, this will give us what execute Time X minus one cube. Close. Um, going to be excellent foreword. And you bring down the three B X minus one squared take to do if it is the inside protester's for one. Um, And then let's simplify this. And yes, expand X minus one squared. If you command it, you get some more complication, and it comes down to X minus one squared time seven x extend. A Ford mine is for X Cube, and we set this whole thing with zero. So we know that for X minus one, we know that every plug in one we get zero. So we know that for one of the values are one and then for seven next to the forth my network cube, because you know we can pull out and execute will be seven. Accident finished. Fourth. It was zero. So now we have and a critical values are X equal one. We already know that from experience. Once word and then we know zero. So plug ins, you know, we get also summon for seven X minus a fourth. If you solve for that, you get poor seventh as well on does our three critical number. So now for the second part. So the second derivative, Kath, what we're going to do is plug in the value. So you first have to take the second derivative. So we take the second derivative, Andi, again, we're going to apply the Ching Teo. Ah, the tropical and I come down to six x word times seven execute minus fifteen X squared plus ten X minus two. Go on, then. Ah, were you going to do is you're going to plug in value so you plug in the critical number. So you plug in. Ah, a primal zero. We got zero. And if we plug in a primary one have double probably won. You also get here then. If you plug in a double crime off four seven, you get approximately. You got approximately two point four and that's good. Indio. So what does this mean? Well, we have some crimes you've got zero. That's an inflection point. So that means that there's a sign change occurring between is in less than members less than zero and after zero. And then another sign change occurred. That one. The number's less than one in greater than one. And then at four seventh we have a positive, which means the Khan came up, which means there's a local men happening. So the second derivative test doesn't exactly tell us where the what to sign changes that are happening. So the only way to do justice you create a sign chart for further evaluation. But are you Khun yet? So you would have to do a sign chart to figure out. Is that where you what are the exact science are shaming? But the second derivative tells us that there is a sign because there is an inflection point, the current and for the is asking us to next question asked of C. What is the first definitive test out with? So when intelligence is that, um, well, we have already taken the first derivative test, so we have to do a signed chart analysis, so we're going to create a sign chart like I've done before. This one's gonna be a little bit bigger. So if we look at it, this three really main component parts of prime. So we have X minus one squared, and we have seven X and fourth planet works. You what? We've breaking down down into execute time seven months before. So we're gonna take it in the piece by piece, uh, part so that we can evaluate it much more easily and quicker. So it would have exploded, Plunge, cleared. That's one of our like a line. And then we have X cube. That's another one of our line. And we have seven x minus four. That's another one of our lives, and we're evaluating it from zip zero fourth for seventh and one. So this is our number line as a reminder again. So for explain this one to squared, we know that anything's whereto have all posits that this will be all positive. And then for X cube, any number less than zero will be negative. And then anything positive will be positive. And then for seven X minus four. If we plug in number of lessons, you know, we get a negative number of a number between German four seven wax, which is one half. You get something like Ah, fourteen for minus sixteen for and that's negative. So this will be negative, a number between four, seven and one, so we can take three fourth and so seven times two is twenty one two, twenty one minus sixteen. That's positive, and this will be positive and then we multiply it to get our sign off. F Prime three negatives have negus positive times. A positive is a positive, positive, positive, positive, negative, negative. And this is all positive and this is a positive. So we know it is increasing, decreasing, increasing and increasing. And what's interesting is here that the fruit derivative test tells us where both the local max and local minutes occurring, whereas the second derivative told us only the local men. So at zero, we know that there's a local Mexican. It is increasing and decreasing, so local max at Europe and then we know that there was a local men at four seventh because it is increasing and then decrease in Crete. I mean, it is decreasing that interesting and that was confirmed from our second derivative toe and We also know the behavior of the function, so we know that is increasing, decreasing and then it increases forever. And the second derivative test doesn't really tell us that. And I feel like the first day of the test that you really valuable. It's a little bit harder to see the con cavity of the graph, but it can be seen through this through the first day of the test, with careful analysis like as we did in this case and yeah, that that is the things that the first derivative and a second round of tests. So there's definitely limitation, but what the second derivative tasks Duncan tell us. But the first three years of tests, all the information is there is just a matter of analysing it and thinking more about the graft. And the second derivative test kind of just confirms what we think about the shape or helps us, you know, have a better picture in our mind about what the exact shape of the graft will be.

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