4.3 Mean Value l heorem www webassign.net/web/Student/Assignment-Responses/submi

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4.3 Mean Value l heorem www webassign.net/web/Student/Assignment-Responses/submit?dep-16662087404 120% Search 4· 0/2 points l Previous Answers SullivanCalcl 4.3.015 My Notes Ask Your Teacher Verify that the function satisfies the three conditions of Rolle's Theorem on the given interval. rx) = sin(2x) on [0, ] This answer has not been graded yet Then find all numbers c in (a, b) guaranteed by Rolle's Theorem. (Enter your answers as a comma-separated list.) Additional Materials eBook 3:35 PM Type here to search 112/2017

Explanation / Answer

As per Rolle's theorem:

If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0

f(x) = sin 2x

verify if f(0) = f(pi) and f(x) is differentiable at both a = 0 and b = pi

sin 0 = sin 2pi => 0 also f(x) is differentiable at both 0 , pi

Thus rolle's theorem applies here and the point c that lies in (0,pi) when f'(c) = 0

f'(c) = 2 cos 2c

0 = cos 2c

c = pi/4 , 3pi/4

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