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(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.$f(\theta)=2 \cos \theta+\cos ^{2} \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$

A. decreasing on $(0, \pi) \quad$ increasing on $(\pi, 2 \pi)$B. $\operatorname{local} \max :(\pi,-1)$C. $\mathrm{CU} :\left(\frac{\pi}{3}, \frac{5 \pi}{3}\right) \quad \mathrm{CD} :\left(0, \frac{\pi}{3}\right),\left(\frac{5 \pi}{3}, 2 \pi\right) \quad$ inflection points $\left(\frac{\pi}{3}, \frac{5}{4}\right),\left(\frac{5 \pi}{3}, \frac{5}{4}\right)$D. SEE GRAPH

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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for this program. We first right out with purity for Jeff Bikini. So for part A, we said F prime Michels a zero. Since that we only consider the domain to be Sillitoe. Mental function reaches from 0 to 2 pi. So for this equation, we only have, ah, very solutions. So X equals zero pie and the two pipes. That means we only have two interval just to concede that the front zero to high in the front pie to to a pipe over the first interval If prime is inactive So the functions depressing over a second life price positives Little function is increasing. Mamie's days are local. Many months at X equals two pi with value F pie e questo minus one For part C. We set a safe on a narrative to be zero. So we have X equals to, um three up high high third, um, pie and the five pi over three. So we have foresight intervals to consider from zero to high or three crump high or three. Too high from pie to five higher with three and the frog five Hire 3 to 2 pipe. So, on the first interval after about promise negative, so the functions can keep down on the second hero. If that book prime years positive solo functions can't give up on the third interval If that book promise Also positive so the families can keep up on the last centavo If double crimes in the active so far looking to convict out. So this all this information way can sketch your graph for if, from from 0 to 2 pi so from 00 hi, you know, to a pipe. So first we notice that at X one X seacoast 20 uh, functioning closed three. So there is a wide intercept here, and the same s when X equals two pi since three because cause I zero he caused to one, um so the fashion is decreasing for us, the increasing the critical point is X equals two pi and this local Milliman We will be, um if pie which is minus one. So the function would go through office three point now based on our reflection. So since we have this conclusion, so we have to inflection points at X equals 23 pilot three and X equals toe pyre. If I so we label that out. So it's higher. Three and the pie, or if I us five pyre or three. So no every due to graft their function is talk about a kind of cave down in the beginning. So when you passed this inflection point, it becomes concave up. Then we have a local minimum and going up. Then we have another inflection point. So it's concave context down again. Well, a about this inflection point and, uh, local ah Minutemen. So this is a graph of F.

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