Please Show all the work step by step for reward. Thank you in advance. DO NOT R

Question

Please Show all the work step by step for reward. Thank you in advance. DO NOT ROUND ANY NUMBERS. All answers must be exact! (I will award extra points if the details are thorough)

Consider the equation below.

(a) Find the interval on which f is increasing. (Enter your answer in interval notation.)


Find the interval on which f is decreasing. (Enter your answer in interval notation.)


(b) Find the local minimum and maximum values of f.

local minimum =

local maximum =

(c) Find the inflection points.

(x, y) = ( , ) (smaller x-value)

(x, y) = ( , ) (larger x-value)

(d) Find the interval on which f is concave up. (Enter your answer in interval notation.)


(e) Find the interval on which f is concave down. (Enter your answer in interval notation.)

local minimum =

local maximum =

Please Show all the work step by step for reward. Thank you in advance. DO NOT ROUND ANY NUMBERS. All answers must be exact! (I will award extra points if the details are thorough) Consider the equation below. f(x) = 8 cos2x - 16 sin x, 0

Explanation / Answer

A. For 0?x?2? let's find the derivative and study it's sign. f'(x)=-16(sin2x+cosx) setting it equal to zero we obtain sin2x+cosx =0 or cosx=0 =>x=pi/2+2pin, x=3pi/2+2pin or 2sinx+1=0=>x=7pi/6+2pin,x=11pi/6+2pin,which has solutions x=pi/2,7pi/6, 3pi/2,,11pi/6 . We then have 3 intervals [pi/2, 7pi/6], [7pi/6 ,3pi/2], [3pi/2, 11pi/6]. If you take test values in each interval, you'll see that the derivative is negative in the middle interval and positive in the other two. Thus f is increasing in [[pi/2, 7pi/6] U [3pi/2, 11pi/6] and decreasing on [7pi/6 ,3pi/2] b. f(7pi/6) = 8cos(7pi/3) -16sin(7pi/6) = =12 and f(3pi/2)=8cos(3pi) -16sin(3pi/2)= 8 the local max value is 12 and the local min. value is 8

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