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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) = \sin x + \cos x $, $ 0 \leqslant x \leqslant 2\pi $

(a) $f(x)=\sin x+\cos x, 0 \leq x \leq 2 \pi, \quad f^{\prime}(x)=\cos x-\sin x=0 \Rightarrow \cos x=\sin x \Rightarrow 1=\frac{\sin x}{\cos x} \Rightarrow$\[\tan x=1 \Rightarrow x=\frac{\pi}{4} \text { or } \frac{\mathrm{sin}}{4} \text { . Thus, } f^{\prime}(x)>0 \Leftrightarrow \cos x-\sin x>0 \Leftrightarrow \cos x>\sin x \Leftrightarrow 0<x<\frac{\pi}{4} \text { or }\]$\frac{5 \pi}{4}<x<2 \pi$ and $f^{\prime}(x)<0 \Leftrightarrow \cos x<\sin x \Leftrightarrow \frac{\pi}{4}<x<\frac{5 \pi}{4} .$ So $f$ is increasing on $\left(0, \frac{\pi}{4}\right)$ and $\left(\frac{5 \pi}{4}, 2 \pi\right)$ and $f$is decreasing on $\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$(b) $f$ changes from increasing to decreasing at $x=\frac{\pi}{4}$ and from decreasing to increasing at $x=\frac{5 \pi}{4} .$ Thus, $f\left(\frac{\pi}{4}\right)=\sqrt{2}$ is alocal maximum value and $f\left(\frac{5 \pi}{4}\right)=-\sqrt{2}$ is a local minimum value.(c) $f^{\prime \prime}(x)=-\sin x-\cos x=0 \Rightarrow-\sin x=\cos x \Rightarrow \tan x=-1 \Rightarrow x=\frac{3 \pi}{4}$ or $\frac{7 \pi}{4} .$ Divide the interval$(0,2 \pi)$ into subintervals with these numbers as endpoints and complete a second derivative chart.

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s so we're being asked to find intervals on which efforts and Christian Duke losing the local maxim men intervals on Khun Cavity and Reflection for FX. So in order to find where after Rex is increasing and decreasing, reply the first derivative cast, which is by first finding the derivative project. And since to do it of F. Beck, I mean to do it of sign of X's Co Sonic and Close A nexus Negative sign, X Oh, we're just going to rewrite it. That's co signed X x minus sign X and just a little side note Here we're working on a very specific interval from zero to two pi his thing from zero to pine. This is very important to know when we're doing our We're trying to figure out where our efforts increasing and decreasing. So then we said our do everything for a zero. So this occurs. We're co signed X equal sign next and at this point, it's very important. It just really know your unit circle arms and so that if you know you're in a circle, you know that thiss sono cosa Mexico Sonics occurs that angles of prior before and five time before innocently something that you have to memorize or no, my heart. So we know that this occurs and be cool. Pi over four and five, Private whore. And so now we're going to apply our sign chart. Um, senses. This is a no functions being exactly multiplied. Would contest right co sign in just right f crime Things were just born to be evaluating the whole function, but we evaluating from zero and then the pilot for the pie over for and then five pile before. Don't bring this over in two parts. No. So since we are being asked to evaluate from zero to two pie away nor any number less than zero and any number of good Ethan to pie And so now the best thing to do is to simply plugging numbers between two. Empire Force of this again requires understanding, unity circle. So you know that the angles last empire forward be part three pi over six. So if you know the values of promise exit complaint the men and if you plug them in, you get you get the sign we'LL be positive and then you do you apply this for each for each values between each number, so this will give negative and positive. So now that we know the signs of prime, we know the sign. We know the intervals on which it is increasing and decreasing, so you can say it is increasing where f promise positive. So there's occurs between zero and to play out front to back your power for and by power for in two parts clear and then it is decreasing. Where I'm so sorry about that on DH, then it is Have the function is decreasing whenever promise lesson dear Oh so this occurs when it is negative and that is occurring between pyre of floor and five prime before push. So now that we know where there's increasing and decreasing, we actually also we could just figure out where the local maxes, max and L and the local minimum is simply by looking at how the function is increasing or decreasing. Just the function is increasing and then decreasing. We know that there's a local max at prior before so private before we have a local Max X equals power for and then then it is increasing and decreasing. So we have a a local men at five before so now and not defined the interval joke on cavity We take the second derivative and we started todo so the second river there is just negative sign X my next cosign X at that equals, you know, bring the co sign over and you take the negative was well, so bring the negative sign over and then does that something really a really neat trick you can do here that make it make your life You can divide signed by co sign and And if you know you're tricking a metric identity sign of the coastline, it's just simply tangent. And then on the right hand, Simon was simply left with negative one. So now we're being asked to find for the slope is the negative one on the unis circle? So where does Tangent X equals negative One dose again requires understanding or knowing the values of the unis trickle. And there's a clear that ex ical three pile before and seven five four. Now that we know where it is equal to zero, we can now apply sign chart. Okay. And this is where our, um it is. These are our values our number line. This will be zero to be a pilot before seven power for and to buy and now we have Tio simply put that plug in the values of F double prime. So again, we also ignore various lesson zero in greater than to pie. So we simply apply would've quietly plugging numbers between Ju and replying to Qianjin Ex Army into f prime So a negative sign in my disco Sanex So if you do these, you get negative positive negative And since these are the signs of f prime, we know that if it's negative, it is Khan came down This positives can't give up and if the negatives can't keep down so it went it so our intervals questions to fill those con cave up looks like a U. This occurs between three piles before and seven pi over four and then con cave down, which is upside down. You occurs between zero and three pi over four and seven. Pilot four and two pi they are inflection. Point occurs when the signs changes for a crime. So this a curve when it was going from negative to positive three pile before and then a shin you again at seven private homes, going from positive to negative. So those are two inflection point, and that is all Thank you.

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