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 using interval notation.)


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


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

(c) Find the inflection points.

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


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

local minimum value     local maximum value 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) = 7 sin x + 7 cos x, 0

Explanation / Answer

For 0?x?2? let's find the derivative and study it's sign. f'(x)=7cosx - 7sinx = 7(cosx -sinx) setting it equal to zero we obtain cosx -sinx =0 or sinx =cosx or sinx/cosx =1 or tanx =1 , which has solutions x=?/4 or x=5?/4 . We then have 3 intervals [0, ?/4], [?/4 ,5?/4], [5?/4, 2?]. 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 [0, ?/4] U [5?/4, 2?] and decreasing on [?/4, 5?/4] b. f(?/4) = 7sin(?/4) + 7cos(?/4) = 7/?2 + 7/?2 = 14/?2 and f(5?/4)=7sin(5?/4)+7cos(5?/4)= -14/?2 the local max value is 14/?2 and the local min. value is -14/?2 c.for the points of inflection , let's get the second derivative f''(x)=-7sinx -7cosx=-7(sinx+cosx) setting to zero , we obtain sinx+ cosx =0 or tanx=-1 meaning x=3?/4 or x=7?/4 The 3 intervals for the second derivative are [0, 3?/4] , [3?/4, 7?/4], [7?/4, 2?] f(3?/4)=7/?2 - 7/?2 =0 the same way f(7?/4)=0 The points of inflection are (3?/4, 0) and (7?/4, 0). If you plug in test values in the 3 intervals, you'll see that the second derivative is negative on the 2 outside intervals ? f is concave down on [0, 3?/4] U [7?/4 ,2?]

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