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Analyzing mistakes when finding extrema (example 1)

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Analyzing the work of someone who tried to find extrema of a function, to see whether they made mistakes.

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  • piceratops ultimate style avatar for user Rowain Hardby
    Isn't this a trick question? Does x^3-6*x^2+12x actually have any relative extrema?
    (2 votes)
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    • leaf green style avatar for user kubleeka
      No, it doesn't. It's derivative is 3x²-12x+12, which factors as 3(x-2)², which is 0 at 2 and positive elsewhere. Since extrema require the derivative to switch between positive and negative, this function has no extrema.
      (7 votes)
  • piceratops ultimate style avatar for user Abhiram
    so any point where the derivative is 0 is called a critical point right??
    (1 vote)
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  • leafers sapling style avatar for user Max  Eastwood
    I'm a little confused. So when you go from a peak at 3, then drop back down to 0, then go back up to 3, that's not a curve? What happens with the line?Isn't the graph you showed a value of -3? Any help would be appreciated!
    (1 vote)
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    • spunky sam orange style avatar for user FlamingArrow
      The key is in the fact that we are working with derivatives. Even though the derivative decreases from 3 to 0 and then back to 3, it never becomes less than zero. Since taking the derivative is just the rate of change/ slope at those points, the slope is always positive over this interval, except for at zero where the tangent line is just horizontal (dy/dx = 0).

      However, if we were taking the actual value of the function and got points (1,3) and (2,0), you are right, the derivate would be negative at any point where 1<x<2.
      (1 vote)
  • duskpin ultimate style avatar for user 3.1415926535897932
    To be more precise, could you just find the one sided limit on both sides of the derivative?
    (1 vote)
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  • sneak peak purple style avatar for user Vector Inc.
    This isn't really related to the topic, but if we had an equation like
    12x^2 + 4x - 4
    Can we factor out a 4x from the terms even though only two of them have it?
    (1 vote)
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Video transcript

- [Instructor] Pamela was asked to find where h of x equals x to the third minus six x squared plus 12 x has a relative extremum. This is her solution. So step one, it looks like she tried to take the derivative. Step two, she tries to find the solution, find where the derivative is equal to zero and she found that it happens at x equals two so she says that's a critical point. In step three, she says, she makes the conclusion that therefore h has a relative extremum there. Is Pamela's work correct? If not, what's her mistake? So pause this video and try to work through it yourself and see if Pamela's work is correct. Alright, well, I'm just gonna try to do it again in parallel. So first, let me just take the derivative here. So h prime of x. Just use the power rule multiple times. It's gonna be three x squared for the x to the third. Two times negative six is negative 12 or minus 12 x and then the derivative of 12 x is plus 12 and let's see. You can factor out a three here. So it's three times x squared minus four x plus four and this part is indeed equal to x minus two squared. So this is equal to three times x minus two squared. So her step one looks right on target. Okay, step two, the solution of h prime of x is equal to zero is equal to x equals two. Yeah, that works out. If you were to say, if you were to say three times x minus two squared which is h prime of x, the first derivative, and set that equal to zero, this is going to be true when x is equal to two and so any point where your first derivative is equal to zero or it's undefined, it is indeed a critical point. So this step looks good so far and then step three, h has a relative extremum at x equals two. Alright, so she made a big conclusion here. She assumed that because the derivative was zero, that we have a relative extremum. So let's just see if you can even just make that conclusion. In order to have a relative extremum, your curve is gonna look something like this and then you would have a relative extremum right over here and over here, your slope goes from being positive and then it hits zero and then it goes to being negative or you could have a relative extremum like this. This would be a maximum point. This would be a minimum point right over here and then in a minimum point, your slope is zero right over there but right before it, your slope was negative and it goes to being positive but you actually have cases where your derivative, your first derivative is zero but you don't have an extremum. So for example, you could have a point like this where right over here, your slope or your derivative could be equal to zero and so your first derivative would be equal to zero but notice, your slope is positive, it hits zero and then it goes back to being positive again and so you can't make the conclusion just because your derivative is zero that it's definitely an extremum. You could say it's a critical point and so step two is correct. In order to make this conclusion, you would have to test what the derivative is doing before that point and after that point and verify that it is switching sides and we could try to do that. So let's make a little table here. Make a little table. Do it a little bit neater. So x, x, h prime of x right over here. We know at x equals two, h prime of two is zero. That's our critical point. But let's try, I don't know, let's see what happens when x is equal to one and then let's see what happens when x equals three. I'm just sampling points on either side of two and let's see, we are going to have, when x is equal to one, h prime of one is three times one minus two squared. One minus two, negative one squared is positive one times three is still positive and then three. Well, three minus two squared times three. That's also gonna be three. So this is actually a situation where, like I've just drawn it where our slope is positive before we hit the critical point, it gets to zero but then it starts becoming positive again and so that's why you actually have to do this test in order to identify whether it's an extremum. It turns out that this is not an extremum. This is not a maximum or a minimum point here. So Pamela's work is not correct and her mistake is step three. In order to make this conclusion, you would have to test on either side of that critical point. Test the first derivative.