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Continuity And Differentiability

Question
CBSEENMA12035506
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Verify Roll's Theorem for the function : straight f left parenthesis straight x right parenthesis equals cos space straight x plus sin space straight x space in space left square bracket 0 comma space 2 straight pi right square bracket

Solution
space Here space straight f left parenthesis straight x right parenthesis space equals space cos space straight x space plus space sin space 2 straight x space space left parenthesis straight a right parenthesis space Since space cos space straight x space and space sin space 2 straight x space are space both space continuous space in space left square bracket 0 comma space 2 space straight pi right square bracket space space therefore space straight f left parenthesis straight x right parenthesis space equals space cos space straight x space plus space sin space straight x space is space continuous space in space left square bracket 0 comma space 2 space straight pi right square bracket space space left parenthesis straight b right parenthesis space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus space sin space straight x space plus space cos space straight x comma space which space exists space in space left parenthesis 0 comma space 2 space straight pi right parenthesis space space therefore space straight f left parenthesis straight x right parenthesis space is space derivable space in space left parenthesis 0 comma space 2 space straight pi right parenthesis space space left parenthesis straight c space right parenthesis space straight f left parenthesis 0 right parenthesis space equals space cos space 0 space plus space sin space 0 space equals space 1 space plus space 0 space equals space 1 space space straight f left parenthesis 2 space straight pi right parenthesis space equals space cos space 2 space straight pi space plus space sin space 2 space straight pi space equals space 1 space plus space 0 space equals space 1 space space therefore space straight f left parenthesis 0 right parenthesis space equals space straight f left parenthesis 2 space straight pi right parenthesis space space therefore space straight f left parenthesis straight x right parenthesis space satisfies space all space the space conditions space of space Rolle apostrophe straight s space Theorem space space therefore space there space must space exist space at space least space one space value space straight c space of space straight x space such space that space straight f apostrophe space left parenthesis straight c right parenthesis space equals space 0 space where space 0 space less than space straight c space less than space 2 space straight pi space space Now space straight f apostrophe left parenthesis straight c right parenthesis space equals space 0 space gives space minus space sin space straight c space plus space cos space straight c space equals space 0 space rightwards double arrow space tan space straight c space equals space 0 space space therefore space straight c space equals space straight pi space element of space left parenthesis 0 comma space 2 space straight pi right parenthesis space space therefore space Rolle apostrophe straight s space Theorem space is space verified.

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