Find the points on the curve y=cosx - 1,in the interval [0,2pi],where the tangent is parallel to x axis.Use Rolles theorem. sir plz give the detail solution.

Asked by bidhanrout65 | 30th Nov, 2014, 08:08: AM

Expert Answer:

R o l l e apostrophe s space T h e o r e m : L e t space f space b e space a space r e a l space v a l u e d space f u n c t i o n space d e f i n e d space o n space t h e space c l o s e d space i n t e r v a l space left square bracket a comma b right square bracket space s u c h space t h a t left parenthesis i right parenthesis space i t space i s space c o n t i n u o u s space o n space t h e space c l o s e d space i n t e r v a l space left square bracket a comma b right square bracket left parenthesis i i right parenthesis space i t space i s space d i f f e r e n t i a b l e space o n space t h e space o p e n space i n t e r v a l space left parenthesis a comma b right parenthesis a n d space left parenthesis i i i right parenthesis space f left parenthesis a right parenthesis equals f left parenthesis b right parenthesis T h e n comma space t h e r e space e x i s t s space a space r e a l space n u m b e r space c element of open parentheses a comma b close parentheses space s u c h space t h a t space f apostrophe left parenthesis c right parenthesis equals 0 G i v e n space f u n c t i o n space i s space f open parentheses x close parentheses equals cos x minus 1 S i n c e space f left parenthesis x right parenthesis space i s space c o n t i n u o u s space o n space open square brackets 0 comma 2 straight pi close square brackets space a n d space d i f f e r e n t i a b l e space o n space open parentheses 0 comma 2 straight pi close parentheses a n d space f open parentheses 0 close parentheses equals cos left parenthesis 0 right parenthesis minus 1 equals 0 equals cos open parentheses 2 straight pi close parentheses minus 1 equals f left parenthesis 2 straight pi right parenthesis comma space we space can space apply space Rolle apostrophe straight s space Theorem. Therefore comma space there space exists space one space real space point space straight c element of open parentheses 0 comma 2 straight pi close parentheses space for space which space straight f apostrophe left parenthesis straight c right parenthesis equals 0 Consider space straight f apostrophe left parenthesis straight c right parenthesis equals minus sinc therefore space straight f apostrophe left parenthesis straight c right parenthesis equals 0 rightwards double arrow minus sinc equals 0 rightwards double arrow straight c equals straight pi rightwards double arrow straight f left parenthesis straight c right parenthesis equals cosπ minus 1 equals minus 2 Thus comma space the space point space on space the space curve space is space left parenthesis straight pi comma minus 2 right parenthesis comma space where space tangent space is space parallel space to space straight x minus axis.

Answered by Vimala Ramamurthy | 30th Nov, 2014, 06:58: PM