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Georgia Southern University

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Problem 56

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)=-2 x+\tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}

$$

Answer

$$

x=\frac{\pi}{4}

$$

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## Discussion

## Video Transcript

alright. Syria FX is negative to X plus ten x on the domain. It's one whole period of tan, so sex between plus minus piratey. I was sick. The derivative finer critical points. So we have negative two plus secret squared ex. Well, when does that prime equals? Zero notice that's seeking is always defined between Native Pirate. Tune by Over two. They don't have ass, intones a at a pie or two. And let's remind Piper, too. But if crime is zero when seeking, Squared is to or second Axe is closer minus Teo well, that's where CO sign his plus or minus one over two. Well, that happens. It pluck, I'll see. So when does that happen? Well, it's only going to happen because of X is only going to be one over two years. So co sign of pirate for is one of her two and co sign of three pyre. For that's outside, the domain is negative. Monitor room too s o the only values here going to be higher for and night before I know the coastline of the size or both. One of you reeked. All right, slumps drone number line put in negative fire too. Cheers. Thank you. Depart before expire. For, however, to let's look at what happens A dysfunction. So it's important to know the pirate for his lesson one see, but bigger than a half. Sophie, look. Att F Crime. Okay, so your native to plus, if we look in between we'll notice that second squared is always going to be home bearing zero. Okay? And let's see so negative by tunic power for So if you look at something like, ah, negative pie over three, So seek it of pirate three is going to be too. So we square it that's going to be forced. The negative two plus four is going to be positive. Okay. And then if we look at seeking of zero, we're going to get once and never to plus one is going to be negative. And again, if you look a higher or three, that's going to see and squared. How every three is going to be force is going to give us a positive value. So it looks like we have a local Max and negative barber for and the value there it's going to be see pie over to minus one and then we have a local men at a pyre for in the value is going to be one Linus Fire two because recall the tan empire before was one and never subtracting two times by diverting power before but just by over two. And so again, if we look at the derivative draft of the function which we should see this coordination between the derivative being positive and the function increasing and then the derivative being negative in the function, decreasing and sing or the derivative changes signs is going to be where we have these local extreme.

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