verify mean value theorem for f(x) = x-2sinx on [-pi,pi] - Maths - Continuity and Differentiability

Question

verify mean value theorem for f(x) = x-2sinx on [-pi,pi]

Rishabh Mittal · Accepted Answer

Dear Student ,

Please find below the solution to the asked query :

fx=x -2 sin x on -π , πi x and 2 sin x both are continous, so fx is continous in -π , πii f'x=1 - 2 cos x exists in -π , πSo , it is derivable
LMT satisfiedThus,fπ-f-ππ--π = f'cπ-2 sin π- -π-2 sin -ππ+π=1-2 cos cπ-2 sin π+π+2 sin -π2π=1-2 cos c2π-2 sin π -2 sin π2π=1-2 cos c       Since sin-x=- sin x2π2π=1-2 cos c       Since sin π = 01=1-2 cos ccos c =0c=±π2     ANS
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