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Problem 63

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.

$$

h(\theta)=3 \cos \frac{\theta}{2}, \quad 0 \leq \theta \leq 2 \pi, \quad \text { at } \theta=0 \text { and } \theta=2 \pi

$$

Answer

$\frac{3}{4}$

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## Discussion

## Video Transcript

So in this problem, we're gonna try to find the local extreme a off a shafia. Each of Peter here is a given function. And we know that, um, theater is gonna be between zero and two pi. So any time we want to find a little cool extreme, um, either and maximal or minimal another function, our first step is to find the derivative. So if we applied the CI rule to, um, Asia feta, we noticed that we're gonna get three. Um, sign off, feed our two times. Um right. And the sign will be negative because the derivative, of course, Chinese minus sign. And we're gonna get times the derivative of the inside function. So in that case, that would be 1/2. Okay, so we see here that then our full derivative is gonna be minus three halfs. Sign Athena over to. But once we find the derivative, how do we know what are the values for the local extreme? Well, we're gonna set, um, that function, uh, are derivative equal to zero and a sol fourth AEA. So remember here, um, we're dealing with actually going to metric functions. There are many ways to do this. But if you notice our equation when we set it to zero here, um, this will amount to be the same. S asking ourselves Went its sign, uh, feta over to equal to zero. Okay. And if you remember the unitary circle, right, Um, sign is zero precisely at, uh, when the argument off our function right is zero or when it is two pi, right? So in our case, this will happen when feta over it too, is equal to zero. Oh, are Yeah. All right, too. Raise is equal to Hi. Some kid. So in that says, um, now we can solve for feta. I'm multiplying. Um, both sides bite Teoh each of those equations. So we get the day. Zero are? Yeah, it was two pi, which is what her problem wanted us to sell. Finally, we must ask yourself, right. Is this a maximum or minimum? And to do that, we can go and grab our function. This is the graph of age off Beatem. Ah, this market here and we see that the maximum happens exactly when theta equals zero and the minimum, right. Uh, let's move this up a little bit. And the minimum happens exactly at two pi, right, recalling that the minimum and the maximum are the Lewis and the, um, highest points of the back respectable.

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