Problem

(a) Find the intervals on which $f$ is increasing…

01:49

Problem 6 Medium Difficulty

(a) Find the intervals on which $f$ is increasing or decreasing.
(b) Find the local maximum and minimum values of $f .$
(c) Find the intervals of concavity and the inflection points.
$f(x)=\cos ^{2} x-2 \sin x, \quad 0 \leqslant x \leqslant 2 \pi$

Answer

A. decreasing on: $\left(0, \frac{\pi}{2}\right),\left(\frac{3 \pi}{2}, 2 \pi\right) \quad$ increasing on: $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
B. The local maximum is $f\left(\frac{3 \pi}{2}\right)=2$
The local minimum is $f\left(\frac{\pi}{2}\right)=-2$
C. $x=\frac{\pi}{6}$ and $x=\frac{5 \pi}{6}$ are the inflection points

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Video Transcript

for this question. We need to compute the derivatives first. So this is a froze. So the purity of office and the second Kira tip off, um, by the product, Roy should be, um, twice off sigh square off X minus, twice off, cause I x squared minus surpassed twice off science. So we need to use the trigger Nidhi to convert this second figurative into something in terms off science. So here it used trick, and it Absolutely, uh, if we do this, we have the second curative equals two two times. Sykes past one. Holmes, Sigh X minus one. Hair or summat. Domain off. If this phone 0 to 2, pi is for only for one period. Okay, So, Patty, we want to fight the increasing, decreasing terraza, which is set a primal cost. Zero. I think that, um, you, in fact, I've Primakov's to minus two times cause i x times brackets, I express one because of zero. So we have co sex because a zero or sigh x past one because it zero in our front zero toe vaccines young toe to pi. We have three solutions XY because two pi have three pie half or X equals. So we only have two solution. Because when X equals 23 pie have both Cossacks and the sex press one or zeros. Okay, so we don't have to Social menus. We have 37 troubles from zero to I have We should use up brackets across the interval here. And if rump I have 23 pie have in a 43 pi have to talk high. Okay, over the first interval. If we can see the first of the narrative is inactive So far she is decreasing the second of all if prime miss positive. So the function is increasing The last interval if price negative. So the function is decreasing again. Which means in part B, we have a local Maybe a month at X equals two pi half with value as pie half. He wants to minus two. Also, we have a local maximum at X equals through a three pie half with Valeo. Uh, if three pie heavy coasted to okay. No, we look at the concave. It is. So, um we said the second figurative because it zero We call that we have a second The narrative here. That's why I factor the second the directive in this way, because on the solution off this equation will be the point. Satisfy science press. Why? Because zero or sigh ex miners 1/2 equals 20 So over this old man from 0 to 2 pi, we actually have three solutions. So X equals two high over a 65 pi over six and the three pie hair. That means we have four different stubby intervals. So first ones from zero to pi over six. Sick wise from pirates 6 to 5 pi over six. They're the wise five higher of a 6 to 3 pie half the last one is three. Pie had to two pac, so we can check. Um, the estaba promise positive or negative over, um, each entero but effect. From here, we can see, uh, the the coefficient to is positive the function of first bracket One process sign hicks. It's known active because the minimum value of sciences minus one. So if amuse one process sign off X is always no negative. That means will need me to take care of the last one. Sigh x minus 1/2. So for the frozen Tebow from the auto pira six. After about promise leg connective them use the function is concave them on a second. Terrible. If double prints positive, the functions can keep up the third interval if taboo price negative again so the functions can keep come. And for the last interval, left double primates again negative, so it's can get them. Okay, So from this result, we only have two inflection points at X equals two, Pirates six and the XY Coastal five PIRA visits because on this day said only two that can have it. He is. The committee changes at that point. Um, X equals 23 pie half is not, uh and the inflection points since, like Unclarity our lesson on both sides off three. Power to the concave it ease our downwards.

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