a. Find the local extrema of each function on the given interval,

and say where they occur.

b. Graph the function and its derivative together. Comment on

the behavior of $f$ in relation to the signs and values of $f^{\prime}. $

$$

f(x)=\frac{x}{2}-2 \sin \frac{x}{2}, \quad 0 \leq x \leq 2 \pi

$$

## Discussion

## Video Transcript

Okay, so you have f of X is sine X Because the next. So the derivative. Yes, Cousin X pleasant eggs. And so when does F prime you call zero? Well, it's when sign ankles and negative Cousin X, or if you prefer Tan X is negative one. Okay, So when does that happen? Well, that happens at three by three. Four in seven pirate for should mention course. We're looking at X values between zero and two pirates for it. So these are the only value is in the central. So we're looking for the local extreme. So if in part here, zero years to pie and then we have repair for you have seven pirate for so we have you don't look a crime. And so if we look at SE something between zero and three pi over four like pie over to that's going to be once a pod. Asai. It's lucky enough Prime. So if prime of power to, he has going to be one just going to give us a positive value. Yep. I we're goingto have negative ones. That's negative. And then, if we look at, say, seven power for maybe negative pie or six. There's eleven, five or six. Well, they're co sign is going to be all right. Three over too. Sign is going to be negative. One half that's going. Digger says it's gonna be positive. Okay, so we have a look, Old Max. Where? Well, hat, Uh, see three pirate for and the value of the function through pirate for is, or two. And then we also have a local max it to pie. The Valium. Negative one. Because we're increasing up to that value. And then we have local men. It okay, zero negative one. Right, Because we're decreasing down from cirrhosis here. Is going to be, are increasing up from zero. Says he was going to be in men and also at seven. By reform in the value there is going to be minus pretty okay, really? Graft the's, the function, the driven. And together we see that we have the local extreme, say, like local Maxima, where the derivative changes science from positive to negative and local minima where the derivative changes signs from negative deposited

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