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# (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.$f(x) = \cos^2 x - 2\sin x$, $0 \leqslant x \leqslant 2\pi$

## a) Decrensing in $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$ Increasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$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) Inflection points at $x=\frac{\pi}{6}$ and $x=\frac{5 \pi}{6}$Concave downward in $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)$Concave upward in $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$

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problem is part a. Find the intervals on which half its increasing our secreting are to be finding the local maximum and minimum values of five. Have C find the intervals off can committee and inflection points. I thought blacks is he goto, assigning square minus two times sighing. Axe Axe is between zero and two pi. It's a four way first to a computer half axe problem, which is equal to who would harm school. Cy Axe. I'm negative kayaks minus two times cause I X, which is the court with two times Kasai axe times elective fine max minus one. If we lie to half axe from you called zero we have X is equal to hi Over too war There is high over two. Then we have the following results. If the interval it's between zero and pi over too, we have extra foam. It's negative. So is Ik fizzy on the interval? Hi Over too two three pile work too. A fax prom is positive. Over half half is increasing and I was an interval. Three Pi over too, too high. So we have affects prom. It's negative. So half is decreasing B from the result in part a half What? Max, It's the call to pie over to you. A fax has a local minimum value, which is the cultural Cyril minus two, which is the cultural negative, too. Axe is equal to three pi over too. If has a maximum value, which is equal to zero plus two is which is equal to two. How to see first of a compute second derivative off half a box, which is equal to who would have negative flying Axe Ham's effective sign Axe minus one class two halves Cose I X Ham's elective course I X, which is equal to two times sine X squire. Plus who would have thought I x minus to co sign X koi. So here we minus two cams. One minus sign. Ex corde, we're half. This is Echo two two times Sign Axe plus one comes to find X minus one. Observation with half What affects prom. Prom. So the second derivative changes size one sign acts is because of you. Half you will have axe is equal to pi over a six or while high over six. So your half into a ll zero two pi over six behalf second derivative is negative. The function come calf downward on into Opie over six to five. High over six. Second derivative. It's positive. The function conch. If up, as in Toronto, file pie over six. You are too high. A secondary motive is selective. The function, Kev. Downward. Yeah.

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