Relative (local) extrema
Analyzing mistakes when finding extrema (example 2)
- [Instructor] Erin was asked to find if f of x is equal to x squared minus one to the 2/3 power has a relative maximum. This is her solution. And then they give us her steps, and at the end they say, is Erin's work correct? If not, what's her mistake? So pause this video and see if you can figure it out yourself. Is Erin correct, or did you she make a mistake, and where was that mistake? All right, now let's just do it together. So she says that this is the derivative. I'm just going to reevaluate here to the right of her work. So let's see, f prime of x is just going to be the chain rule. I'm gonna take the derivative of the outside with respect to the inside. So this is going to be 2/3 times x squared minus one to the 2/3 minus one, so to the negative 1/3 power, times the derivative of the inside with respect to x. So the derivative of x squared minus one with respect to x is two x. (siren ringing) There's a fire hydrant, a fire (laughing), not a hydrant, that would be a noisy hydrant. There's a fire truck outside, but okay, I think it's passed. But this looks like what she got for the derivative. Because if you multiply two times two x, you do indeed get four x. You have this three right over here in the denominator. And x squared minus one to the negative 1/3, that's the same thing as x squared minus one to the 1/3 in the denominator, which is the same thing as the cubed root of x squared minus one. So all of this is looking good. That is indeed the derivative. Step two, the critical point is x equals zero. So let's see, a critical point is where our first derivative is either equal to zero or it is undefined. And so it does indeed seem that f prime of zero is going to be four times zero, it's gonna be zero over three times the cubed root of zero minus one, of negative one. And so this is three times negative one, or zero over negative three, so this is indeed equal to zero. So this is true. A critical point is at x equals zero. But a question is, is this the only critical point? Well as we've mentioned, a critical point is where a function's derivative is either equal to zero or it's undefined. This is the only one where the derivative's equal to zero, but can you find some x-values where the derivative is undefined? Well what if we make the derivative, what would make the denominator of the derivative equal to zero? Well if x squared minus one is equal to zero, you take the cube root of zero, you're gonna get zero in the denominator. So what would make x squared minus one equal to zero? Well x is equal to plus or minus one. These are also critical points because they make f prime of x undefined. So I'm not feeling good about step two. It is true that a critical point is x equals zero, but it is not the only critical point. So I would put that there. And the reason why it's important, you might say, "Well what's the harm in not noticing "these other critical points? "She identified one, "maybe this is the relative maximum point." But as we talked about in other videos, in order to use the first derivative test, so to speak, and find this place where the first derivative is zero, in order to test whether it is a maximum or minimum point, is you have to sample values on either side of it to make sure that you have a change, a change in sign of the derivative. But you have to make sure that when you test on either side that you're not going beyond another critical point. Because critical points are places where you can change direction. And so let's see what she does in step three right over here. Well it is indeed in step three that's she's testing, she's trying to test values on either side of the critical point that she, that the one critical point that she identified. But the problem here, the reason why this is a little shady, is this is beyond another critical point that is less than zero, and this is beyond, this is greater than another critical point that is greater than zero. This is larger than the critical point one, and this is less than the critical point negative one. What she should've tried is x equals 0.5 and x equals negative 0.5. So this is what she shoulda done is try maybe negative two, negative one, negative 1/2, zero, 1/2, and then one we know is undefined, and then positive two. Because this is a candidate extremum, this is a candidate extremum, and this is a candidate extremum right over here. And so you wanna see in which of these situations you have a sign change of the derivative. And you just wanna test in the intervals between the extremum points. So I would say that really the main mistake she made is in step two is not identifying all of the critical points.