Answer the following questions about the functions whose derivatives are given. a. What are the critical points of $f ?$ b. On what open intervals is $f$ increasing or decreasing? c. At what points, if any, does $f$ assume local maximum and minimum values? $$ f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi $$

All right. So here we have some trig functions. It's our derivative is sine X plus goes on X times signed x, my escorts and necks Okay, which I can actually rewrite a sine squared x minus consents for Dex. Okay. And so if I want to know where f crime is equal to zero, well, that's going to be when sine squared of X is you go to co sign squared of X. When that happens when side of X is either plus or minus co sign of X, right, Because we have to take a square roots, we get maps of value. So you have to consider the positive or negative this happens when X is, um since this is your day, Yes. So we know that sign X is going to be equal to co sign X. Now we're only considering values. Ah, between zero and two pi. So we know pyre before three point four five five one seven times before sine x and cosign x. We're going to have the same ninety two. Hi. Go over for three. Fire for five. Fire before hand. Seven tire for right. So there are critical points and so Let's play. He's on a graph. So Yeah. So one thing we can do is we can say this craft is just pi times accent. And this isthe one fourth. This is three for us, five fourths. This is seven. Four. So all this is saying is that I'm just instead of burying pie every time this number line is just, uh hi Times X. Okay. So if I go down, say to nanyan test value mes in between these guys. So let's look at the ground. So, you know, one good thing to be able to check is maybe at half values by something. Check it half what we have over here. Okay? Not meal to check half over there. So maybe a six and then with the other half, man. So we contest it one and then at three halves, say it. Eleven, six of nine or six. Okay, so we know he's a little bigger and from zero kind of these at these points were just testing in between. So one sixth are we really just need to know which is bigger. Signer Coastline s O at one sixth co sign. It's going to be bigger. That's gonna be negative prior to this is actually going to be zero. So it's just gonna be sign. It's just gonna be one. So it's positive. And then at one, this is going to be zero, and this is just going to be minus one on, So that's going to be negative. And then three halves again, this is going to be a sign of three pi over to you is gonna be one. It's going negative one. That square is going to be one on divine, the serious a positive and again over here because I was going to be bigger. So it's negative. So then we see that is decreasing an increasing decreasing, then increasing, decreasing, decreasing between hope zero and pi over four and then increasing between higher for and three pipe before and decrease saying between through fire for and five fire for increasing between five fiver for seven fiver for and then finally decreasing between seven. Fire for and to buy. Okay, so we have lots of extreme out here. Let's listen in. So where do we have a local men? Well, that's when f changes from being decreasing to increasing. We have local men's at X equals looks like pyre for and five, five or four and then local. Max is Cat X equals so, but it's a change from being increasing and decreasing three pirate fuller and some buyers before.

## Discussion

## Video Transcript

All right. So here we have some trig functions. It's our derivative is sine X plus goes on X times signed x, my escorts and necks Okay, which I can actually rewrite a sine squared x minus consents for Dex. Okay. And so if I want to know where f crime is equal to zero, well, that's going to be when sine squared of X is you go to co sign squared of X. When that happens when side of X is either plus or minus co sign of X, right, Because we have to take a square roots, we get maps of value. So you have to consider the positive or negative this happens when X is, um since this is your day, Yes. So we know that sign X is going to be equal to co sign X. Now we're only considering values. Ah, between zero and two pi. So we know pyre before three point four five five one seven times before sine x and cosign x. We're going to have the same ninety two. Hi. Go over for three. Fire for five. Fire before hand. Seven tire for right. So there are critical points and so Let's play. He's on a graph. So Yeah. So one thing we can do is we can say this craft is just pi times accent. And this isthe one fourth. This is three for us, five fourths. This is seven. Four. So all this is saying is that I'm just instead of burying pie every time this number line is just, uh hi Times X. Okay. So if I go down, say to nanyan test value mes in between these guys. So let's look at the ground. So, you know, one good thing to be able to check is maybe at half values by something. Check it half what we have over here. Okay? Not meal to check half over there. So maybe a six and then with the other half, man. So we contest it one and then at three halves, say it. Eleven, six of nine or six. Okay, so we know he's a little bigger and from zero kind of these at these points were just testing in between. So one sixth are we really just need to know which is bigger. Signer Coastline s O at one sixth co sign. It's going to be bigger. That's gonna be negative prior to this is actually going to be zero. So it's just gonna be sign. It's just gonna be one. So it's positive. And then at one, this is going to be zero, and this is just going to be minus one on, So that's going to be negative. And then three halves again, this is going to be a sign of three pi over to you is gonna be one. It's going negative one. That square is going to be one on divine, the serious a positive and again over here because I was going to be bigger. So it's negative. So then we see that is decreasing an increasing decreasing, then increasing, decreasing, decreasing between hope zero and pi over four and then increasing between higher for and three pipe before and decrease saying between through fire for and five fire for increasing between five fiver for seven fiver for and then finally decreasing between seven. Fire for and to buy. Okay, so we have lots of extreme out here. Let's listen in. So where do we have a local men? Well, that's when f changes from being decreasing to increasing. We have local men's at X equals looks like pyre for and five, five or four and then local. Max is Cat X equals so, but it's a change from being increasing and decreasing three pirate fuller and some buyers before.

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