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Continuity And Differentiability

Question
CBSEENMA12035508
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Verify Roll's Theorem for the function :straight f left parenthesis straight x right parenthesis equals space 4 space sin space straight x space in space left square bracket 0 comma space straight pi right square bracket.

Solution
straight f left parenthesis straight x right parenthesis space equals space 4 space sin space straight x space space left parenthesis straight a right parenthesis space space space space Since space sine space curve space is space continuous space space therefore space straight f left parenthesis straight x right parenthesis space is space straight a space continuous space function space in space left square bracket 0 comma space straight pi right square bracket space space left parenthesis straight b right parenthesis space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 space cos space straight x comma space which space exists space in space left parenthesis 0 comma 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 straight pi right parenthesis. space space left parenthesis straight c right parenthesis space space space space straight f left parenthesis 0 right parenthesis space equals space 4 space sin space 0 space equals space 0 space comma space straight f left parenthesis straight pi right parenthesis space equals space 4 space sin space straight pi space equals space 0 space space therefore space straight f left parenthesis 0 right parenthesis space equals space straight f left parenthesis 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 in space left parenthesis 0 comma space straight pi right parenthesis space such space that space straight f apostrophe left parenthesis straight c right parenthesis space equals space 0. Now space straight f apostrophe left parenthesis straight c right parenthesis equals 0 space gives space us space 4 space cos space straight c equals 0 comma space straight i. straight e. comma space cos space straight c equals 0 therefore space straight c equals straight pi over 2 comma space fraction numerator 3 straight pi over denominator 2 end fraction comma space fraction numerator 5 straight pi over denominator 2 end fraction comma... space space space rightwards double arrow space straight c equals straight pi over 2 space lies space in space left parenthesis 0 comma space straight pi right parenthesis Rolle apostrophe straight s space theorem space is space verified.

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