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Determine the intervals on which the function is concave up and intervals on which the...

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cspanutius | Student, Undergraduate | Salutatorian

Posted July 17, 2012 at 10:40 PM via web

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Determine the intervals on which the function is concave up and intervals on which the function is concave down. f(x)=x- 2sin x defined on the interval (0, 3 pi)

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beckden | High School Teacher | (Level 1) Educator

Posted July 18, 2012 at 1:25 AM (Answer #1)

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f(x) is concave up when f''(x) > 0 and concave down when f''(x) < 0

f(x) = x-2sin(x)

f'(x) = 1-2cos(x)

f''(x) = 2sin(x)

f''(x) > 0 when sin(x) > 0 and on the interval (0,3pi) this happens on the intervals (0,pi) and (2pi,3pi).

f''(x) < 0 when sin(x) < 0 and this happens on the interval (pi,2pi).  So our answer is

f(x) is concave up on the intervals (0,pi) and (2pi,3pi), and the function is concave down on (pi,2pi).

We can verfy with a graph.

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